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MODÜL MATEMATİK - 2 Alt bölümlerin Karma Testler ÜSTEL VE LOGARİTMİK EDĜOñNODUñQñL©HULU Üstel ve Logaritmik Fonksiyonlar KARMA TEST - 1 Modülün sonunda FONKSİYONLAR tüm alt bölümleri 1. loga16 = 4 5. f^xh = log2 9log3^3x - 2hC L©HUHQNDUPDWHVWOHU PMEVôVOBHÌSF G-1 LBÀUS \\HUDOñU PMEVôVOBHÌSF MPH8BLBÀUS ³ Üstel Fonksiyon t 2 A) 2 3 C) 1 D) 1 E) 1 A) 35 83 67 D) 17 E) 11 B) 3 2 B) C) 3 24 3 3 ³ Logaritma Fonksiyonu t 4 ³ Logaritmanın Özellikleri t 9 ³ Taban Değiştirme Kuralı t 15 ³ Logaritma Fonksiyonunun Grafiği t 17 6. log 16 . log 125 . log 49 = log 4 x ^8xh 2. log2x = 0,68 57 8 6ñQñIð©LðĜOH\\LĜ PMEVôVOBHÌSF 17 x50 JGBEFTJOJOFöJUJLBÀUS PMEVôVOBHÌSF YLBÀUS %XE¸O¾PGHNL¸UQHN A) 2 B) 4 C) 6 D) 8 E) 16 ³ Üstel Denklemler t 23 VRUXODUñQ©¸]¾POHULQH A) 2 B) 3 C) 4 D) 5 E) 8 ·/÷7&34÷5&:&)\";*3-*, 3. MODÜL ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 www.aydinyayinlari.com.tr ³ Logaritmalı Denklemler t 24 ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3* ³ Üstel ve Logar·iTtUmFMa'PlıOLETşJZiPtOsizlikler t 31 ÖRNEK 3 G3Z3+ G Y = - -a2 + a + Y TANIM GPOLTJZPOVÐTUFMCJSGPOLTJZPOEVS ³ Üstel ve Logaritm BikTBFZoTnkEFsOiGyBoSLMnQlPa[rJUJlGaSFFMTBZPMTVO BOOBMBCJMFDFôJFOHFOJöEFôFSBSBMôOFEJS İlgili Gerçek Hayat Problemleri t 33 G 3 Z 3+ G Y = aY õFLMJOEF UBONMBOBO 3. f_ x i = log2_ x + 3 i 7. log2 = x, log3 = y, log7 = z <HQL1HVLO6RUXODU GPOLTJZPOMBSBÑTUFMGPOLTJZPOEFOJS LVSBM JMF WFSJMFO G GPOLTJZPOVOVO FO HFOJö UB- PMEVôVOBHÌSF MPHOJOY Z [UÑSÑOEFOFöJUJ 0RG¾O¾QJHQHOLQGH\\RUXP BTBZTOBÐTUFMGPOLTJZPOVOUBCBO YEFóJõLF- ONLÑNFTJBöBôEBLJMFSEFOIBOHJTJEJS BöBôEBLJMFSEFOIBOHJTJEJS \\DSPDDQDOL]HWPHYE OJOFüsEFOJS EHFHULOHUL¸O©HQNXUJXOX <(1m1(6m/6258/$5B) (-2, R) VRUXODUD\\HUYHULOPLĜWLU ³ Karma Testler t 39 A) ( -3, -2 ) C) ( -R, -3 ) A) x + y + z B) 3x - y + z C) 2xÜ+ste2lyv+e zLogaritmik Fonksiyonlar $\\UñFDPRG¾OVRQXQGD WDPDPñ\\HQLQHVLOVRUXODUGDQ D) ( -3, -2 ] E) [ -2, R) D) 2x - 2y + z E) x - 3y + 5z ROXĜDQWHVWOHUEXOXQXU ³ Yeni NeÖsRilNESKor1ular t 47 DNñOOñWDKWDX\\JXODPDVñQGDQ 1. Kayakla atlama yapan bir sporcunun rampadan ay- 3. 1MBKEB CVMVOBO LVN UBOFDJLMFSJOJO NN DJOTJOEFO \"öBôEBLJ GPOLTJZPOMBSO IBOHJMFSJOJO ÑTUFM GPOLTJ- XODĜDELOLUVLQL] ZPOPMEVôVOVCFMJSMFZJOJ[ SMELUBOTPOSBZBUBZEPóSVMUVEBBMEóIFSYNNF- WFSJMFOPSUBMBNB¿BQE FóJNJNPMNBLÐ[FSF TBGFEF[FNJOFEFóEJóJOPLUBZBEÐõFZEPóSVMUVEBLJ a) G Y = 2Y ÖRNEK 4 V[BLMóG Y PMNBLÐ[FSF m = MPHE+ G Y =2 -Y + 2 ÑTUFMGPOLTJZPOEVS GPOLTJZPOVOBHÌSF G EFôFSJLBÀUS b) f (x) = ^ 2 hx 1 G Y = log2 -Y CBóOUTWBSES ÑTUFMGPOLTJZPOEVS x 4. f_ x i = 23x + 1 PMBSBLNPEFMMFONJõUJS p c) f(x) = f 1 ÑTUFMGPOLTJZPOEVS y8. 1 + 1 1MBKEBOBMOBOLVNUBOFDJLMFSJOJO¿BQMBSOOPSUBMB- ÑTUFMGPOLTJZPOEFôJMEJS 1 NBT NNPMBSBLIFTBQMBONõUS 2 PMEVôVOBHÌSF G-1 Y BöBôEBLJMFSEFOIBOHJTJ- 1- 1 1 - EJS logab logba E G Y = -2 Y e) G Y = -2Y A) log2x + 1 B) log3x + 1 C) log3x - 1 JöMFNJOJOTPOVDVBöBôEBLJ#MFVSOEBFOHÌISBFO HCJVTJEQJMSBKOFôJNJLBÀUS 2 2 2 ÑTUFMGPOLTJZPOEVS A) loga b - logba B) l(ologga(0a,0b6) , -1,2) f) G Y = -2-Y ÑTUFMGPOLTJZPOEVS ÖRNEK 5 D) log2x + 1 E) log2x - 1 C) logzaefmiban p x D) \"a # $ G Y = B- Y -ÐTUFMGPOLTJZPOVWFSJMJZPS 3 3 E) 1 & H G Y =Y G = $OW%¸O¾P7HVWOHUL O b FöJUMJôJOFHÌSF f ( - EFôFSJLBÀUS ÑTUFMGPOLTJZPOEFôJMEJS % I G Y =Y2 TEST - 1 ÑTUFMGPOLTJZPOEFôJMEJS Üstel Fonksiyon Buna göre, 39 5. B 6. A 7. C 8. E 1. D 2. B 3. E 4. E * :BUBZEPóSVMUVEBNFUSFZPMBMBOTQPSDVEÐ- 1. f : R Z R+UBONMBOBO –x + 1 5. 1 p õFZEPóSVMUVEBNZPMBMNõUS f(x) = f I. f ( x ) = 2–x 4 ÖRNEK 2 ** ;FNJOEFONFUSFZÐLTFLMJLUFZLFOZBUBZEPó- G3Z3+ G Y = - B-II. Y f ( x ) =Õx GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ SVMUVEBNFUSFZPMBMNõUS III. f ( x ) = 1x PMBCJMJS GPOLTJZPOVÑTUFMGPOLTJZPOPMEVôVOBHÌSF BOOFO A) y B) y C) y Her alt bölümün *** 3BNQBEBO BZSMEó BOEBO [FNJOF UFNBTOB VRQXQGDRE¸O¾POHLOJLOL LBEBSEÐõFZEPóSVMUVEBNFUSFZPMBMNõUS HFOJö EFôFSBSBMôOFEJS GPOLTJZPOMBSOEBO IBOHJMFSJ ÑTUFM GPOLTJZPO- EVS ÖRNEK 6 x JGBEFMFSJOEFOIBOHJMFSJEPôSVEVS \" :BMO[* # :BMO[**G3Z$3 +* WF*G* Y = 3Y- + x x E) y D) y \" :BMO[* # :BMO[** $ :BMO[*** GP%O L*T*JWZFPO**V* OB HÌSF f ( 2x + GPOLTJZPOVOVO G Y % *WF*** % *WF** & **WF*** UÑSÑOEFOFöJUJOFEJS 4. #JSCËMHFEFCVMVOBOJOTBOTBZTÐTTFMZËOUFNJMF xx IFTBQMBOSLFO P JML OÐGVT S ZMML PSUBMBNB OÐGVT BSUõ I[ U JML OÐGVT IFTBCOEBO TPOSB HF¿FO [B- 2. f : R Z R+ , f ( x ) = ( 3n - 6 )x-1 WHVWOHU\\HUDOñU 2. #JS¿FLJSHFOJOCJSJODJT¿SBZõOEBMPHNFUSF NBO UZMTPOSBQMBOMBOBOOÐGVTQPMNBLÐ[FSF JLJODJ T¿SBZõOEB MPH NFUSF пÐODÐ T¿SBZ- GPOLTJZPOV ÑTUFM GPOLTJZPO PMEVôVOB HÌSF O 6. f ( x ) = 4 . 73x - 1 õOEBMPHNFUSFPMBDBLõFLJMEFOT¿SBZõO- P = PFSU OJOFOHFOJöEFôFSBSBMôBöBôEBLJMFSEFOIBO- EBBMEóZPMG O PMNBLÐ[FSF CJ¿JNJOEFNPEFMMFOJS HJTJEJS PMEVôVOBHÌSF G -1 LBÀUS G O = log O+ \"OLBSBhOO ZMOEBLJ OÐGVTV PMVQ 1. a, b, c, e, f 2. ^ 3\",3 3h \\ ( 7 2 # 3= *2 7 43. ( –1,$2 ) \\-( Þ1 -2 5 1+ 5 2 4. 11 15 6. 9.f2 ( x )–18 f ( x )+10 PMBSBLNPEFMMFONJõUJS PSUBMBNBOÐGVTBSUõI[ZÐ[EF PMBSBLIFTBQ- 2 3 2 5. - MBONõUS , 4\" 4 # $ D) 2 E) 1 D) _ 2, 3 i \\ * 7 4 & Þ 3 7. y #VOB HÌSF \"OLBSBhOO ZMOEB CFLMFOFO g( x ) = bx OÑGVTVZBLMBöLPMBSBLLBÀUS (e0,36 = 1,4) h( x ) = cx #V ÀFLJSHF LF[ TÀSBELUBO TPOSB CBöMBEô OPLUBEBOen fazlaLBÀNFUSFV[BLMBöNöUS 3. f (x) = 2x + 2x + 2 2x + 2 - 2x + 1 f( x ) = ax \" # $ x \" # $ % & % & :VLBSEBWFSJMFOG Y H Y WFI Y GPOLTJZPO- PMEVôVOBHÌSF G EFôFSJLBÀUS MBSOBHÌSF B CWFDTBZMBSOOTSBMBNBTBöB- ôEBLJMFSEFOIBOHJTJEJS A) 5 # 5 $ % 4 E) 3 1. E 2. D 48 3. D 4. A 2 4 3 2 A) a < b <D # B< c <C $ C< a < c 4. f^ x h = 1 D) b < c < a E) c < b < a 10x 8. y GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ f( x ) = ( a + 1 )x + b PMBCJMJS A) y B) y C) y 1 1 4 x x 2 D) y E) y 1x x O1 :VLBSEB HSBGJôJ WFSJMFO G Y = ( a + 1 )Y + b 1x x GPOLTJZPOVOBHÌSF f ( a +C EFôFSJLBÀUS A) 6 # $ % & 1. C 2. D 3. A 4. A 6 5. B 6. E 7. C 8. E
www.aydinyayinlari.com.tr ·/÷7&34÷5&:&)\";*3-*, ÜNİVERSİTEYE HAZIRLIK 3. MODÜL MATEMATİK - 2 ÜSTEL VE LOGARİTMİK FONKSİYONLAR ³ Üstel Fonksiyon t 2 ³ Logaritma Fonksiyonu t 4 ³ Logaritmanın Özellikleri t 9 ³ Taban Değiştirme Kuralı t 15 ³ Logaritma Fonksiyonunun Grafiği t 17 ³ Üstel Denklemler t 23 ³ Logaritmalı Denklemler t 24 ³ Üstel ve Logaritmalı Eşitsizlikler t 31 ³ Üstel ve Logaritmik Fonksiyonlarla İlgili Gerçek Hayat Problemleri t 33 ³ Karma Testler t 39 ³ Yeni Nesil Sorular t 47 1
·/÷7&34÷5&:&)\";*3-*, 3. MODÜL ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 www.aydinyayinlari.com.tr ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3* ·TUFM'POLTJZPO ÖRNEK 3 TANIM G3Z3+ G Y = - -a2 + a + Y GPOLTJZPOVÐTUFMCJSGPOLTJZPOEVS BTBZTEFOGBSLMQP[JUJGSFFMTBZPMTVO BOOBMBCJMFDFôJFOHFOJöEFôFSBSBMôOFEJS G 3 Z 3+ G Y = aY õFLMJOEF UBONMBOBO -a2 + a +â WF -a2 + a + 2 > 0 GPOLTJZPOMBSBÑTUFMGPOLTJZPOEFOJS a2 - a -â WF -a + 2) (a + 1) > 0 BTBZTOBÐTUFMGPOLTJZPOVOUBCBO YEFóJõLF- WF D = 1 + 4 = –1 2 OJOFüsEFOJS a≠ 1± 5 WF – +– 1,2 2 ÖRNEK 1 ^ - 1, 2 h \\ * 1 - 5 1+ 54 , 22 \"öBôEBLJ GPOLTJZPOMBSO IBOHJMFSJOJO ÑTUFM GPOLTJ- ZPOPMEVôVOVCFMJSMFZJOJ[ a) G Y = 2Y ÑTUFMGPOLTJZPOEVS ÖRNEK 4 b) f (x) = ^ 2 hx ÑTUFMGPOLTJZPOEVS G Y =2 -Y + 2 ÑTUFMGPOLTJZPOEVS x ÑTUFMGPOLTJZPOEFôJMEJS GPOLTJZPOVOBHÌSF G EFôFSJLBÀUS c) f(x) = f 1 p 2 E G Y = -2 Y f(4) = 3 . 22 - 4 + 2 = 3 · 1 + 2 = 11 44 e) G Y = -2Y ÑTUFMGPOLTJZPOEVS f) G Y = -2-Y ÑTUFMGPOLTJZPOEVS ÖRNEK 5 H G Y =Y ÑTUFMGPOLTJZPOEFôJMEJS G Y = B- Y -ÐTUFMGPOLTJZPOVWFSJMJZPS G = I G Y =Y2 ÑTUFMGPOLTJZPOEFôJMEJS FöJUMJôJOFHÌSF f ( - EFôFSJLBÀUS f ( 2 ) =JTF B- 2)2 - 4 = 12 ( a - 2 )2 = 16 ÖRNEK 2 a - 2 =WFZBB- 2 = -4 G3Z3+ G Y = - B- Y ·TUFMGPOLTJZPOPMEVôVOEBOUBCBOOFHBUJGPMBNB[ GPOLTJZPOVÑTUFMGPOLTJZPOPMEVôVOBHÌSF BOOFO f ( x ) = 4x - 4 j f (-1) = 1 15 HFOJö EFôFSBSBMôOFEJS -4 =- 44 2a -â WF B- 6 > 0 ÖRNEK 6 Bâ WF B> 6 WF B> 3 G3Z3+ G Y = 3Y- + 7 (3, Þ) \\ ( 7 2 a≠ GPOLTJZPOVOB HÌSF f ( 2x + GPOLTJZPOVOVO G Y 2 UÑSÑOEFOFöJUJOFEJS 2 3 x f (x) = 3 + 1 & x = 3^ f (x) - 1 h 3 f (2x + 1) = 2x + 1a x = 3. (f (x) - 1 k yaz›l›r. 3 3 = 7 3.^ f (x) - 1 hA 2 + 1 = 2 (x) - 18 f (x) + 10 9f 1. a, b, c, e, f 2. ^ 3, 3 h \\ ( 7 2 2 3. ( –1, 2 ) \\ ( 1 - 5 1+ 5 2 4. 11 15 6. 9.f2 ( x )–18 f ( x )+10 2 , 5. - 22 44
www.aydinyayinlari.com.tr ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 3. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 7 ÖRNEK 10 G Y = 3Y- 2 G3Z3+ G Y =Y + GPOLTJZPOVOVO HÌSÑOUÑTÑOÑO f 1 , 19 p BSBMô GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ 91 x = - JÀJO f^ - 2 h = 11 JÀJOEFCVMVOBOLBÀUBOFUBNTBZEFôFSJWBSES 9 1 < x – 2 < 19 y x = - JÀJO f^ - 1 h = 5 7 3 91 3 3 x = JÀJOG = 3 x - 2 j {-4, -3, -2, -1, 0, 1, 2} 5/3 x x = JÀJOG = x j { -2, -1, 0, 1, 2, 3, 4 }UBOF 11/9 1 –2 –1 O 1 ÖRNEK 8 ÖRNEK 11 G Y = 36Y H Y = 3Y+ G3Z3+ G Y =-Y GPOLTJZPOMBSOEBG Y =H Y FöJUMJôJOFHÌSF 4xJGB- EFTJOJOEFôFSJLBÀUS GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ 36x = 32x . 3 y x = -JÀJOG -2 ) = 12 9x . 22x = 32x . 3 12 x = -JÀJOf ( -1 ) = 6 4x = 3 6 x = JÀJOG = 3 24x = ( 4x )2 = 32 = 9 3 3/2 3 –2 –1 O 1 x = JÀJOG = 2 x ÖRNEK 9 SONUÇ G3Z3+ G Y = 3Y G Y = aYÐTUFMGPOLTJZPOVOEBB`3+ - {} J¿JO GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ J B JTFGBSUBO JJ B JTFGB[BMBOES x = -JÀJO f^ - 2 h = 1 ±SOFóJO y 9 y x = -JÀJO f^ - 1 h = 1 y 3 2 2 1 1 3 x = JÀJOG = 1 1 x –1 x 1 x x = JÀJOG = 3 1/3 y = 2x 1 x 1/9 (Artan) 2 y= –2 –1 O 1 (Azalan) 8. 9 3
·/÷7&34÷5&:&)\";*3-*, 3. MODÜL ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 www.aydinyayinlari.com.tr ÖRNEK 12 -PHBSJUNB'POLTJZPOV \"öBôEBLJ fonksiyonlardan hangileri artan fonksiyondur? 7$1,0%m/*m a) G Y = 3Y- 9 BSUBOES G3Z3+WFB`3+ \\ {}J¿JO 3 –x + 1 B[BMBOES G Y = aY ÐTUFM GPOLTJZPOV CJSF CJS WF ËSUFO BSUBOES b) f (x) = f p GPOLTJZPOEVS 2 G3+ Z3WFB`3+ \\ {}WFY>J¿JO G- Y =MPHaYGPOLTJZPOVOBBUBCBOOBHÌSF c) f (x) = ^ 2 hx MPHBSJUNBGPOLTJZPOVEFOJS E G Y = Õ -Y B[BMBOES y = aY lY=MPHay B[BMBOES e) f(x) = f 1 x BSUBOES 3 p -2 f) G Y = -Y - 2 ÖRNEK 13 %m/*m y G Y = aYJMFG- Y =MPHaY CJSCJSJOFHËSFUFSTGPOLTJZPOPMEVóVJ¿JOZ=Y EPóSVTVOBHËSFTJNFUSJLUJS 5 a > 1 ise 0 < a < 1 ise f( x ) = 2ax + b 2 y f( x ) = ax y O2 f( x ) = ax x xx ,PPSEJOBUEÑ[MFNEFLJG Y = 2ax +CPMEVôVOBHÌSF y=x f–1( x ) = logax y=x f–1( x ) = log x a +CUPQMBNLBÀUS a f ( 0 ) = 2 j 20 + b = 2 j b = 1 ÖRNEK 15 f ( 2 ) = 5 j 22a + 1 = 5 j a = 1 j a + b = 2 G3Z3+ G Y = 3Y+ - 2 ÖRNEK 14 PMEVôVOBHÌSF G–1 Y OFEJS y Z = 3x + 1 - 2 Z+ 2 = 3x + 1 6 x + 1 =MPH3 Z+ 2) j x =MPH3 Z+ 2) - 1 f-1(x) =MPH3(x + 2) - 1 3 y = ( a – 2 )–x + b ÖRNEK 16 –2 O x G3+ Z3 G Y = 2 -MPH3 Y+ ,PPSEJOBUEÑ[MFNEFLJG Y = ( a - 2 )-x +CGPOLTJZP- PMEVôVOBHÌSF f-1 Y OFEJS OVOBHÌSF G LBÀUS Z= 2 -MPH3(x + 1) jMPH3(x + 1) = 2 -Z f ( 0 ) = JÀJO B- 2 )0 + b = 3 j b = 2 j x + 1 = 32 -Z j x = 32-Z - 1 j f-1(x) = 32-x - 1 f ( -2 ) =JÀJO B- 2 )2 + 2 = 6 ( a - 2 )2 = 4 j |a - 2| = 2 a - 2 >PMBDBôOEBOB- 2 =EJS f(x) = d 1 x f^ 2 h = 9 n +2j 22 12. a, c, f 9 4 15. MPH3(x + 2) – 1 16. 32–x – 1 13. 2 14. 2
www.aydinyayinlari.com.tr ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 3. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 17 ÖRNEK 21 G3Z3+ , G Y =Y+ f ^xh = log^1– x2h^x2 + 3x + 7h PMEVôVOBHÌSF G-1 LBÀUS GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJOJCVMVOV[ 2 . 52x + 1 = 250 1 - x2 >WF-x2âWFY2 + 3x +> 0 52x + 1 = 125 2x + 1 = 3 j x = 1 x = -WFY= 1 x2â 3 ÖRNEK 18 –1 1 Yâ f : (1, Þ) Z3UBONM – +– G Y = 3 +MPH2 Y- GPOLTJZPOVOBHÌSF G-1 LBÀUS ( -1, 1 ) \\ { 0 } Z= 3 +MPH2( x - 1 ) jZ- 3 =MPH2(x - 1) ÖRNEK 22 j x - 1 = 2Z-3 j x = 2Z-3 +1 j f-1( x ) = 2x-3 + 1 j f-1( 6 ) = 9 G Y =MPH2 Y2 - L- Y+ GPOLTJZPOV IFS Y SFFM EFôFSJ JÀJO UBONM CJS GPOLTJ- %m/*m ZPOJTFLOJOEFôFSBSBMôOCVMVOV[ G Y =MPHN Y O Y GPOLTJZPOVOVOUBONMPMB- x2 - L- 2 ) x + 1 > 0 –2 2 CJMNFTJJ¿JON Y > WFN Y áWFO Y > 0 D = L-2 )2 - 4 < 0 + –+ PMNBMES L2 -L+ 4 - 4 < 0 ÖRNEK 19 (-2, 2 ) G Y =MPH Y- -Y GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJOJCVMVOV[ x -1 >WFY-âWF- x > 0 ÖRNEK 23 x >WFYâWFY< 5 ( 1, 5 ) \\ { 2 } G Y =MO Y2 - L- Y+L- ÖRNEK 20 GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJ3={ a }PMEV- f (x) = log c 7-x m ôVOBHÌSF B+LLBÀUS ^x – 2h x + 5 x2 - L- 1 ) x +L- 2 =UBNLBSFPMNBMES Ô= 0 ) GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJOFEJS Ô= L- 1 )2 - L- 2 ) = 0 L2 -L+ 9 = 0 x - 2 >WFY-âWF 7-x >0 L- 3 )2 = 0 L= 3 x+5 x2 - 2x + 1 = 0 j ( x - 1 )2 = 0 j x = 1 = a L+ a = 4 x >WFYâWF –5 7 – +– ={3} 1 18. 9 19. (1, 5) \\ {2} 20. =\\^ 5 21. (–1, 1) \\ {0} 22. (–2, 2) 23. 4
TEST - 1 ·TUFM'POLTJZPO 1. f : R Z R+UBONMBOBO 5. f(x) = f 1 –x + 1 * G Y = 2mY 4 p ** G Y =ÕY *** G Y =Y GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ PMBCJMJS A) y B) y C) y GPOLTJZPOMBSOEBO IBOHJMFSJ ÑTUFM GPOLTJZPO- EVS x x x D) y E) y \" :BMO[* # :BMO[** $ *WF** % *WF*** % **WF*** xx 2. G3Z3+ G Y = O- Y- GPOLTJZPOV ÑTUFM GPOLTJZPO PMEVôVOB HÌSF O 6. G Y =Y- OJOFOHFOJöEFôFSBSBMôBöBôEBLJMFSEFOIBO- PMEVôVOBHÌSF G -1 LBÀUS HJTJEJS \" 3 # 3=* 7 4 $ -Þ \" # $ % & 3 % _ 2, 3 i \\ * 7 4 & Þ 3 y g( x ) = bx h( x ) = cx 3. f (x) = 2x + 2x + 2 f( x ) = ax x 2x + 2 - 2x + 1 PMEVôVOBHÌSF G EFôFSJLBÀUS \" 5 # 5 $ % 4 & 3 :VLBSEBWFSJMFOG Y H Y WFI Y GPOLTJZPO- 2 4 3 2 MBSOBHÌSF B CWFDTBZMBSOOTSBMBNBTBöB- ôEBLJMFSEFOIBOHJTJEJS \" B<C<D # B< c <C $ C< a < c 4. f^ x h = 1 % C< c <B & D<C< a 10x 8. y GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ f( x ) = ( a + 1 )x + b PMBCJMJS A) y B) y C) y 1 1 4 x x 2 D) y E) y 1x x O1 :VLBSEB HSBGJôJ WFSJMFO G Y = ( a + 1 )x + b 1x x GPOLTJZPOVOBHÌSF f ( a +C EFôFSJLBÀUS \" # $ % & 1. C 2. D 3. A 4. A 6 5. B 6. E C 8. E
·TUFM'POLTJZPO-PHBSJUNB'POLTJZPOV TEST - 2 1. ,PPSEJOBUEÐ[MFNEFLJG Y =B C+ -Y +DGPOL- 5. MPH3 Y+ = 4 TJZPOVOVOHSBGJóJWFSJMNJõUJS FöJUMJôJOJTBôMBZBOYEFôFSJLBÀUS y \" # $ % & 3 x 2 1 O1 #VOBHÌSF G - LBÀUS \" # $ % & 2. G Y = 2Y+ + 4 6. MPH2 Y2 -Y+ = 3 FöJUMJôJOJ TBôMBZBO Y EFôFSMFSJOJO ÀBSQN LBÀ- US GPOLTJZPOV JÀJO # f ( x ) < FöJUTJ[MJôJOJ \" -6 # - $ - % & TBôMBZBOLBÀGBSLMYUBNTBZEFôFSJWBSES \" # $ % & 3. G3Z3+UBONMBOBOÐTUFMGPOLTJZPOMBSWFSJMNJõUJS #VOBHÌSF * f (x) = f 5 x 13 p MPH Y+ = 2 ** f(x) = c π x FöJUMJôJOJ TBôMBZBO Y EFôFSMFSJOJO ÀBSQN LBÀ- US 4 m *** G Y = -Y \" - # - $ % & GPOLTJZPOMBSOEBO IBOHJMFSJ BSUBO GPOLTJZPO- EVS \" :BMO[* # :BMO[** $ :BMO[*** % *WF** & *WF*** 4. G3Z3+ G Y =Y- + 8. MPH2 MPH3 Y- = 2 PMEVôVOBHÌSF G-1 LBÀUS FöJUMJôJOJTBôMBZBOYEFôFSJLBÀUS \" # $ % & \" # $ % & 1. B 2. B 3. A 4. C 5. E 6. A C 8. B
TEST - 3 -PHBSJUNB'POLTJZPOV 1. MPH Y+ Y+ = 2 5. f^ x h = log f x + 2 p FöJUMJôJOJTBôMBZBOYEFôFSMFSJOJOUPQMBNLBÀ- 3-x US GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJBöBôEB- \" - # - $ % & LJMFSEFOIBOHJTJEJS \" -Þ - # -Þ $ - Þ % -Þ - b Þ & - 2. f :f - 1 , 3 p \" R, 1 - 1 WF ÌSUFO GPOLTJZPO PM- 6. G Y =MPH Y- Y2 +Y 4 NBLÑ[FSF G Y =MPH Y+ GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJBöBôEB- PMEVôVOBHÌSF G -1 LBÀUS LJMFSEFOIBOHJTJEJS \" Þ - { 2 } # Þ \" # $ % & $ - % Þ - { 2 } & 3. G Y = 2Y+ 3 H Y =MPH3 Y+ f^ x h = 1 - log^ x - 2 h PMEVôVOBHÌSF GPH-1) ( 2 ) LBÀUS GPOLTJZPOVOVOFOHFOJöUBO NLÑNFTJOEFLBÀ UBNTBZEFôFSJWBSES \" # $ \" # $ % & % & 8. f ^xh = log –1 ^x + 1h 5 4. G Y =MPH2YWFH Y =MPHY GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJBöBôEB- LJMFSEFOIBOHJTJEJS PMEVôVOB HÌSF GPH JGBEFTJO JO EFôFSJ \" -Þ - # - > $ -4, - LBÀUS \" # $ % & % -3, - & 1. E 2. B 3. D 4. B 8 5. E 6. D B 8. B
www.aydinyayinlari.com.tr ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 3. MODÜL ·/÷7&34÷5&:&)\";*3-*, ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3** 0OMVLWF%PôBM-PHBSJUNB'POLTJZPOV q=(//m.a TANIM a `3+ / { } C > O N`3WFNáPMTVO 5BCBO PMBO MPHBSJUNB GPOLTJZPOVOB POMVL MPHaCO =OpMPHaC MPHBSJUNBGPOLTJZPOVWFZBCBZBôMPHBSJUNB GPOLTJZPOVEFOJS 1 MPHaNC= m pMPHaC G3+ A3 G Y =MPHYWFZBG Y =MPH Y CJ¿JNJOEFHËTUFSJMJS MPHaNCO = n m pMPHaC F= PMNBLÐ[FSF UBCBOFPMBO ÖRNEK 3 MPHBSJUNBGPOLTJZPOVOBEPôBMMPHBSJUNBGPOL- TJZPOVEFOJS log 3 clog 4 4 8 mJöMFNJOJOTPOVDVLBÀUS G3+ A3 G Y =MPHFYWFZBG Y =*O Y CJ¿JNJOEFHËTUFSJMJS = log d log a 3/2 k n ÖRNEK 1 1/2 1/2 2 3 2 MPH +MOY = b l b l FöJUMJôJOJTBôMBZBOYEFôFSJLBÀUS 3/2 +*OY= 10 = 2 pMPH3 d 1/2 log 2 n *OY= 3 2 x = e3 =pMPH3 3 = 2 -PHBSJUNB'POLTJZPOVOVO²[FMMJLMFSJ q=(//m.a ÖRNEK 4 a !3+ / { } MPHa=WF*O= 0 log 1000 + log 2 MPHaa =WF*OF= 8 JGBEFTJOJOFöJUJLBÀUS ÖRNEK 2 ln c 1 m + log 9 e 27 log 7 + 5 ln e 3 = 1 10 7 log 10 + logb 23 l2 3+ 3 = = 20 3 log 1 + log 10 3 2 -1/2 a 2 k 12 1 Ine + log –+ JGBEFTJOJOFöJUJLBÀUS 3 3 23 6 b l 1+ 5 3 =6 ÖRNEK 5 0 +1 log 5 3 25 4 125 JGBEFTJOJOFöJUJLBÀUS 5 log 2·3·4 a 5 3 k4 · a 2 k4 · 5 3 5 5 24 12 8 3 log 5 · 5 · 5 5 24 12 8 3 log 5 · 5 · 5 5 24 23 log 5 5 23/24 23 23 log 5 = · log 5 = 5 24 5 24 1. e3 2. 6 9 23 3. 2 4. 20 5. 24
·/÷7&34÷5&:&)\";*3-*, 3. MODÜL ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 www.aydinyayinlari.com.tr q=(//m.a ÖRNEK 9 a !3+ / { } C>WFD>PMTVO 1 · log 36 - 1 · log 27 - 1 · log 16 234 MPHa CpD =MPHaC+MPHac JGBEFTJOJOFöJUJLBÀUS MPHa c b m =MPHaC-MPHac log^ 36 h1/2 - log^ 27 h1/3 - log^ 16 h1/4 c =MPH-MPH-MPH =MPHd 6 n =MPH= 0 ÖRNEK 6 3·2 MPH2Y= a, MPH2y =C MPH2[= c ÖRNEK 10 FöJUMJLMFSJOF HÌSF log f x2 · y3 p JGBEFTJOJO B C D z +MPH23 +pMPH2 2 JGBEFTJOJOFöJUJLBÀUS UÑSÑOEFOFöJUJOFEJS MPH22 +MPH23 +MPH2125 MPH2(2 · 3 · 125) 23 MPH2 log f x ·y p =MPH2x2 +MPH2Z3 -MPH2[ 2z =MPH2x +MPH2Z-MPH2[ = 2a + 3b - c ÖRNEK 7 ÖRNEK 11 *O Y *O ZWF*O [ MPH224 +MPH2-MPH2-MPH24 PMEVôVOBHÌSF *OJOY Z [UÑSÑOEFOEFôFSJOFEJS JöMFNJOJOTPOVDVLBÀUS x =*O=*O3 · 32 · 5 =*O3 +*O2 +*O MPH2d 24·48 n =MPH22 = 1 x =*O+*O+*O 144·4 x =Z+[+*O *O= x -Z-[ ÖRNEK 12 log x = 1, 3 _ bb log y = 0, 5 ` ÖRNEK 8 bb log z = 3, 2 a *O+*O-*OJGBEFTJOJOFöJUJLBÀUS PMEVôVOBHÌSF YZ[EFôFSJLBÀUS *O3 +*O-*O2 In 8 · 5 = Ind 40 n MPHY+MPHZ+MPH[= 1,3 + 0,5 + 3,2 99 MPH YpZp[ = 5 YpZp[= 105 6. 2a + 3b – c YmZm[ 8. lnd 40 n 10 9. 0 10. MPH2 11. 1 12. 105 9
www.aydinyayinlari.com.tr ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 3. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 13 ÖRNEK 17 log5 e 6 o + log5 c 7 m + log5 c 8 m+ . . . + log5 c n+1 m= 2 MPH TJOY TJOY +MPH TJOY DPTY 5 6 7 n JöMFNJOJOTPOVDVLBÀUS PMEVôVOBHÌSF OEFôFSJLBÀUS MPH TJOY TJOYpDPTY =MPH TJOY TJOY = 1 log d 6 · 7 · 8 ·...· n+1 n=2 5 6 7 n log d n + 1 n = 2 & n + 1 = 25 5 5 O+ 1 = 125 jO= 124 ÖRNEK 14 ÖRNEK 18 MPH= MPH= MPH d 18 + log 1 3 n JGBEFTJOJOFöJUJLBÀUS PMEVôVOBHÌSF MPHLBÀUS 3 MPH p =MPH+MPH2 10 log 1 3 = log 1 3 =-2 =MPHf p +pMPH – 2 32 =MPH-MPH+pMPH = 1 - 0,301 +p = 1,653 3 MPH8( 18 - 2 ) MPH23 24 = log 3 ^ 24 h = 4 2 3 b l ÖRNEK 19 log x + log 1 y + 2 log z = log z2 - log y 22 2 2 2 ÖRNEK 15 FöJUMJôJOJTBôMBZBOYEFôFSJLBÀUS MPH= a log x - log y + 2 log z = log 2 - log y PMEVôVOB HÌSF MPH49! JGBEFTJOJO B UÑSÑOEFO FöJ- 22 22 2 UJOFEJS z MPH49! =MPH( 49 · 48! ) 22 =MPH49 +MPH48! log f x.z p = log f z p =MPH2 +MPH48! = 2 + a 2y 2y 22 x.z z y = y &x=1 ÖRNEK 20 ÖRNEK 16 log ^7x + 2yh - 1 = log ^x + yh 55 FöJUMJôJOFHÌSF x PSBOLBÀUS y 2 · log ^2 7– 3 3h ^2 7 + 3 3h + log ^6 + 35h ^6 - 35h log ^ 7x + 2y h - log ^ x + y h = 1 55 JöMFNJOJOTPOVDVLBÀUS log f 7x + 2y p=1 5 x+y 2· log 1 ^ 2 7 + 3 3 h + log 1 ^ 6– 35 h 7x + 2y fp fp = 5 & 7x + 2y = 5x + 5y x+y 2 7+3 3 6– 35 2x = 3y - 2 - 1 = -3 x3 y=2 13. 124 14. 1,653 15. a + 2 16. –3 11 1 4 19. 1 3 18. 20. 3 2
TEST - 4 -PHBSJUNB'POLTJZPOVOVO²[FMMJLMFSJ 1. log 6 + 4 log 49 + 3 · Ine17 5. log c 2 m + log c 49 m - log c 7 m 57 58 54 JöMFNJOJOTPOVDVLBÀUS JöMFNJOJOTPOVDVLBÀUS 1 1 3 & \" # $ % & \" # $ % 4 2 2 6. MPH+MPH-MPHf 1 p 7 2. log ^ln e25h + lnc 1 log ^9ehm JöMFNJOJOTPOVDVLBÀUS 5 23 JöMFNJOJOTPOVDVLBÀUS \" # $ % & 1 # $ % 7 \" & 2 2 log x = 1 22 log2y = 1 3 3. MPHY= 1 PMEVôVOBHÌSF log 3 x 4 x JGBEFTJOJOFöJUJLBÀ- log2z = 6 US PMEVôVOBHÌSF YpZp[ÀBSQNLBÀUS \" # $ % & 1 11 & \" # $ % 4 32 4. log 3 clog4 4 8 m 8. MPH=B MPH=CPMNBLÐ[FSF log ^0, 25h MPH JO B WF C UÑSÑOEFO FöJUJ BöBôEBLJMFSEFO IBOHJTJEJS 4 \" C+ a + # B+C+ $ B+C- JöMFNJOJOTPOVDVLBÀUS % & % C- a + & C- a + \" - # - $ 1. B 2. D 3. D 4. A 12 5. A 6. B E 8. E
-PHBSJUNB'POLTJZPOVOVO²[FMMJLMFSJ TEST - 5 1. Y Z! R+PMNBLÑ[FSF 5. log f a .b2 p * MPH YZ =MPHY+MPHZ c JGBEFTJBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS ** MPH Y+Z =MPHYMPHZ \" 2 log a + 1 log b + log c *** logf x p = log x - log y 2 y # 1 log a + 2 log b + log c log x 2 *7 log^ x - y h = $ 1 log a + 2 log b - log c log y 2 7 log x = log x - log y % 2 log a + 1 log b - log c log y 2 & 1 log a + 1 log b - log c FöJUMJLMFSJOEFOLBÀUBOFTJEBJNBEPôSVEVS 22 \" # $ % & 2. log 5 6. log 2 a log 625 k 25 5 JGBEFTJOJOFöJUJLBÀUS \" 1 # 1 JGBEFTJOJOTPOVDVLBÀUS 4 2 $ % & \" # 2 $ % 2 2 & 3. log 3 a = 3 log 8523 4 2 2 PMEVôVOBHÌSF BLBÀUS JGBEFTJOJOTPOVDVLBÀUS \" # $ \" 5 # 8 $ 5 % 5 & 9 & 4 5 3 2 2 % 4. log 1 8 8. MPHY=PMEVôVOBHÌSF 32 log 3 x2 x x JGBEFTJOJOFöJUJLBÀUS JGBEFTJOJOFöJUJLBÀUS \" # $ % & \" - 3 # - 1 $ 2 % 3 & 2 5 25 4 3 1. B 2. A 3. E 4. A 13 5. C 6. E C 8. C
TEST - 6 -PHBSJUNB'POLTJZPOVOVO²[FMMJLMFSJ 3 x2 . y .z _ log log c x m = 3a bb 1. xyz 5. y` JGBEFTJBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS - log^x.yh = abba \" 2 log x + 1 log y + 1 log z PMEVôVOB HÌSF Y BöBôEBLJMFSEFO IBOHJTJOF 322 FöJUUJS \" -2a # -a $ # 11 log x - 1 log y - 1 log z % a & 2 - a 3 32 $ - 1 log x + 1 log y - 1 log z 632 % 5 log x + 1 log y + 1 log z 662 & 1 log x - 1 log y + 1 log z 662 6. ln ax2. y3k = 74 ln ax. y2k = 4 2. log f x2. y p + log f z3 p - log f z2 p PMEVôVOB HÌSF Y BöBôEBLJMFSEFO IBOHJTJOF z x3. y x FöJUUJS JGBEFTJOJOFöJUJOFEJS \" F # F2 $ F3 % F4 & F \" log f z3 p # MPH YZ[ $ log c x m yz x2 % & MPH= WFMPH= PMNBLÑ[FSF MPH JGBE FTJOJOEFôFSJLBÀUS 3. MPH= PMNBLÐ[FSF \" # $ MPHJGBEFTJLBÀUS % & \" # $ % & 8. MPH =Y 4. MPH=BPMNBLÐ[FSF PMEVôVOB HÌSF MPH5 JGBEFTJOJO EFôFSJ BöBôEBLJMFSEFOIBOHJTJOFFöJUUJS MPH JGBEFTJ BöBôEBLJMFSEFO IBOHJTJOF FöJU- UJS \" -B # -B $ -B \" x + 1 # Y+ $ Y+ 2 2 & Y+ 2 % -B & B % Y 1. E 2. D 3. D 4. C 14 5. D 6. B A 8. C
www.aydinyayinlari.com.tr ·45&-7&-0(\"3÷5.÷,'0/,÷4:0/-\"3 3. MODÜL ·/÷7&34÷5&:&)\";*3-*, ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3*** 5BCBO%FôJöUJSNF,VSBM ÖRNEK 4 q=(//m.a MPH=BWFMPH=C PMEVôVOB HÌSF MPH JGBEFTJOJO B WF C UÑSÑOEFO a, c `3+ / { } C>PMTVO FöJUJOFEJS MPHaC= logcb MPH=MPH pp logca =MPH+MPH2 +MPH = b + 2a + 1 ÖRNEK 1 ÖRNEK 5 log 27 MPH3 =BWFMPH2 =C 4 PMEVôVOB HÌSF MPH12 JGBEFTJOJO B WF C UÑSÑOEFO FöJUJOFEJS log 3 4 JGBEFTJOJOFöJUJLBÀUS MPH3=MPH3 ( 33 ) =pMPH33 = 3 log 42 log ^ 3 · 2 · 7 h 77 log 42 = = 12 log 12 log a 22 · 3 k ÖRNEK 2 7 7 log 4 8 log 5 256 log 3 + log 2 + log 7 777 JGBEFTJOJOFöJUJLBÀUS = 2 log + log 3 2 77 a+b+1 = 2b + a log5 4 8 = log 2 3/4 = 3/4 log 2 256 8/5 8/5 2 2 15 ÖRNEK 6 = MPH2=BWFMPH23 =C 32 PMEVôVOB HÌSF MPH JGBEFTJOJO B WF C UÑSÑOEFO FöJUJOFEJS ÖRNEK 3 log 16 3 MPH9 + log 90 log a 5 · 2 · 2 k log 12 22 3 3 = JGBEFTJOJOFöJUJLBÀUS log 10 log ^ 5 · 2 h 22 2 log 5 + log 3 + log 2 22 2 = MPH129 +MPH1216 MPH12( 9·16 ) log 5 + log 2 MPH12( 144 ) = 2 22 a + 2b + 1 = a+1 1. 3 2. 15 3. 2 15 4. 2a + b + 1 a +b + 1 a + 2b + 1 5. 6. 32 a + 2b a+1
·/÷7&34÷5&:&)\";*3-*, 3. MODÜL ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 www.aydinyayinlari.com.tr q=(//m.a q=(//m.a B C`3+ / { }PMNBLÐ[FSF B C D`3+ / { }WFE`3+ PMNBLÐ[FSF MPHaCpMPHCDpMPHcE=MPHaE MPHaC= 1 log a ÖRNEK 10 b MPHplog 5 7pMPHpMPH49 10 ÖRNEK 7 JGBEFTJOJOFöJUJLBÀUS 1 +1 logb 3 a 2 k. logb 1/2 l7. log 4 logb 2 a 1/2 k log 216 log 216 2 5 7 l 5 2. l 10 24 9 =d 2 · log 5 n · a 2 · log 7 k · ^ 4 · log 2 h · d 1 · log 10 n JGBEFTJOJOFöJUJLBÀUS 3 4 2 5 7 MPH21624 +MPH2169 =MPH216( 24 · 9 ) =MPH216216 = 1 21 4 = d · 2 · 4 · n · log 5 · log 7 · log 2 · log 10 = 3 4 25 73 ÖRNEK 8 ÖRNEK 11 1 ,ËõFHFOV[VOMVLMBSlog 2 7DNWFMPHDNPMBOEJL- 1+ 1 EËSUHFOJOLËõFHFOMFSJCJSCJSJOFEJLUJS #VOBHÌSF EJLEÌSUHFOJOBMBOLBÀDN2EJS log 3 2 JGBEFTJOJOFöJUJOFEJS log 7 · log 16 log(21/2)7 · log7 ^ 24 h \"-\"/ = 27 = 1 11 22 = = = log 3 6 1+ log 2 log 3 + log 2 log 6 2 log 7 · 4 · log 2 3 33 3 27 = =4 2 ÖRNEK 12 A \"#$пHFO ÖRNEK 9 log 2 log95 [ AD ]B¿PSUBZ 3 MPH2 =B MPH2 =C | |\"# =MPH32 PMEVôVOB HÌSF MPH35 JGBEFTJOJO B WF C UÑSÑOEFO | |\"$ =MPH9 FöJUJOFEJS | |%$ =MPH2 B x D log 5 C 2 log 14 log ^ 2 · 7 h log 2 + log 7 22 22 | |:VLBSEBLJWFSJMFSFHÌSF BD =YLBÀUS log 14 = = = 35 log 35 log ^ 5.7 h log 5 + log 7 22 22 log 2 log 5 3 = 9 & log 2. log 5 = x. log 5 1+ 1 b+1 x log 5 32 9 b b b+1 2 == = 1 ab + 1 ab + 1 a+ & log 2. log 5. log 9 = x bb 325 x=2 1 8. MPH63 b+1 16 4 11. 4 12. 2 9. 10. ab + 1 3
www.aydinyayinlari.com.tr ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 3. MODÜL ·/÷7&34÷5&:&)\";*3-*, q=(//m.a ÖRNEK 17 a `3+ / { }WFC D`3 PMNBLÐ[FSF G Y =*OYGPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ CMPHac = cMPHaC 1 JÀJOGd 1 y aMPHaC =C e e 2 x= n = -1 1 ÖRNEK 13 x =JÀJOG = 0 1/e x –1 1 e2 F*O +MPH- 9MPH32 e x =FJÀJOG F = 1 JGBEFTJOJOFöJUJLBÀUS x = e2JÀJOG F2 ) = 2 5*OF +MPH3 - 2MPH39 5 + 3 - 22 = 4 ÖRNEK 14 ÖRNEK 18 1 G Y =MPHYGPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ log 1000 4 8 + e log3e JGBEFTJOJOFöJUJLBÀUS 1 1 y JÀJOGd 1000MPH84 + e*O x= 22 n =1 log 3 b 2 2 l (2 ) 1000 + e ln 3 x =JÀJOG = 0 1 2 x =JÀJOG F = -1 –1 1/2 1 2 4x 3· –2 10 3 + 3 = 103 x =JÀJOG = -2 ÖRNEK 15 1 + log f – Inf 1 log 2 JGBEFTJOJOFöJUJLBÀUS %m/*m 2 pp 10 e + 9 4 G Y =MPHaYGPOLTJZPOVOVOHSBGJóJ 1 log b – ln –2 l log 2 a > 1 iken 0 < a < 1 iken e 4 y y 10 .10 + 9 1 10 · ( 2 ) + 9 f 20 + 3 = 23 1 1a x x -PHBSJUNB'POLTJZPOVO(SBGJôJ \"SUBOES a1 f ÖRNEK 16 \"[BMBOES G Y =MPH3YGPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ G Y = aYGPOLTJZPOVOVOHSBGJóJ a > 1 iken 0 < a < 1 iken y y f 1 x= 1 JÀJOG d 1 n = -1 y af 3 3 2 a 1x 1 \"[BMBOES x =JÀJOG = 0 1 1/3 1 1 3 9x \"SUBOES x x =JÀJOG = 1 –1 x =JÀJOG = 2 13. 4 14. 103 15. 23
·/÷7&34÷5&:&)\";*3-*, 3. MODÜL ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 www.aydinyayinlari.com.tr ÖRNEK 19 y=logax ÖRNEK 22 y = f( x ) y y y=logcx y=logdx x 2 x y=logbx O 11 9 Yukarıda verilen grafikteki logaritma fonksiyonlarına õFLJMEFWFSJMFOG Y =MPHaYGPOLTJZPOVOVOHSBGJôJOF göre a, b, c, d sayılarını küçükten büyüğe doğru sı- HÌSF G–1( - LBÀUS ralayınız. 1 11 b < a < c <E ff p = 2 & log f p = 2 & a = 9 a9 3 f^ x h = log x & f–1 (x) = d 1 x & f–1 (- 1) = d 1 –1 (1/3) 3 n 3 n =3 ÖRNEK 20 y 1f –3 x ÖRNEK 23 –2 O MPHTBZTOOEFôFSBSBMôOFEJS õFLJMEFWFSJMFOf ( x ) =MPHa ( bx + c ) GPOLTJZPO VOVO 102 < 625 < 103 HSBGJôJOFHÌSF B+ b +DUPQMBNLBÀUS MPH2 <MPH<MPH3 2 <MPH< 3 j (2, 3) x = -JÀJO-3b + c = 0 x=-JÀJOMPHa (-2b + c ) = 0 j -2b + c = 1 b =WFD= 3 x =JÀJOMPHac = 1 j a = c = 3 j a + b + c = ÖRNEK 21 ÖRNEK 24 G3A3 G Y = 2 +MPH3 Y- a =MPH3 C=MPH2WFD=MPH GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ TBZMBSOOLÑÀÑLUFOCÑZÑôFTSBMBOöOCVMVOV[ x-1>0x>1 y 33 << 34 MPH333 <MPH3<MPH334 3 <MPH3< 4 4 24 < 19 < 25 MPH224 <MPH219 <MPH225 4 <MPH219 < 5 x =JÀJOG = 2 3 52 < 122 < 53 MPH552 <MPH5122 <MPH553 2 <MPH5122 < 3 x =JÀJOG = 3 2 c<a<b x =JÀJOG = 4 12 4 x 10 19. C B D E 20. 18 22. 3 23. (2, 3) 24. c < a < b
-PHBSJUNB'POLTJZPOVOEB5BCBO%FôJöUJSNF,VSBM TEST - 7 1. log 625 5. MPH2=YPMNBLÑ[FSF log 1 MPH 24 125 JGBEFTJBöBôE BLJMFSEFOIBOHJTJOFFöJUUJS JGBEFTJOJOFöJUJLBÀUS \" - 4 # - 2 $ - 1 2 4 A x + 3 # x + 9 $ 2x + 1 % & 2x + 1 2x + 3 x+3 3 3 33 3 % 2x + 1 & x + 5 x+1 2x + 5 log 27 6. MPH=YWFMPH=ZPMNBLÑ[FSF 2. 4 + In8 + log log 25 MPH 20 log 3 In4 2 5 2 JöMFNJOJOFöJUJLBÀUS JGBE FTJBöBôE BLJMFSEFOIBOHJTJOFFöJUUJS \" # $ % & y 2-y 2-y \" # $ 3 - 2y 3 - 2x 2 + x - 2y x+y y-x+1 % & 2y - x 2x - y + 2 3. *O=B *O=CWF*O= c MPH= WFMPH= PMNBLÑ[FSF PMEVôVOB HÌSF MPH15 p F OJO B C D UÑSÑO- MPH EFOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS JGBEFTJBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS \" 2a + b + 1 # a + 2b + 1 $ a + b + 1 b+c a+b a+c % 2a + 2b + 2 & 2a + 2b + 1 \" # $ b+c b+c % & 4. A \"#$пHFO 8. MPH=B MPH=CPMNBLÐ[FSF 49 6 [ ED ] // [#$] MPHOJOBWFCDJOTJOEFOEFôFSJOFEJS log 6 D | |E AD å=MPH6DN \" B+C # B+C $ B+C B | |x #%å=MPH6DN log6 7 | |C AE å=DN % B+C & B+C | | :VLBSEBLJ WFSJMFOMFSF HÌSF EC æ = Y LBÀ DN 9. loga x2.y k x3 = 3 4 EJS \" # $ PMEVôVOBHÌSF MPHxZEFôFSJLBÀUS % MPH2 & MPH29 \" # $ % & 1. A 2. D 3. E 4. C 19 5. B 6. C C 8. E 9. B
TEST - 8 -PHBSJUNB'POLTJZPOVOEB5BCBO%FôJöUJSNF,VSBM 1. MPH=BPMNBLÐ[FSF MPH5 BöBôEBLJMFS- 5. 1 + 1 + . . . + 1 log 17! log 17! log 17! EFOIBOHJTJOFFöJUUJS 23 17 \" a - 1 # 1 - a $ a + 1 JöMFNJOJOTPOVDVLBÀUS a a a \" # $ % & & 2a + 1 % 2a - 1 a a 2. logx c a m= 2 6. 6 + 2 b 2 + log 6 1+ log 8 23 JöMFNJOJOTPOVDVLBÀUS logx c b m= 3 \" # $ % & c logx c c m= 4 d PMEVôVOB HÌSF MPHxa - MPHxE JGBEFTJOJO FöJUJ LBÀUS \" # $ % & logxa = 1 , logya = 1 , logza = 1 4 3 2 PMEVôVOBHÌSF MPHa YZ[ EFôFSJLBÀUS \" # $ % & 3. MPH2=YPMEVôVOBHÌSF, MPH2( 62! + 63! + 64! ) JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS \" Y+ # Y+ $ Y+ & Y+ % Y+ 4. 1 + 1 + 1 8. 1+ 2 log 30 log 30 log 30 1+ 1 log 2 235 3 JöMFNJOJOTPOVDVLBÀUS JöMFNJOJO FO TBEF CJÀJNJ BöBôEBLJMFSEFO IBO- HJTJEJS \" MPH4 # MPH4 $ MPH6 & MPH624 \" # $ % & % MPH6 1. A 2. B 3. C 4. A 20 5. A 6. B D 8. E
-PHBSJUNB'POLTJZPOVOEB5BCBO%FôJöUJSNF,VSBM TEST - 9 1. f 1 log 25 6. log 25. log 3 2 . log 7 64 p8 27 35 JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS JöMFNJOJOTPOVDVLBÀUS A 1 # 1 $ 1 % 1 & \" # 4 $ 8 % & 5 625 3 3 2 125 25 5 ,FOBSV[VOMVLMBSMPH5DNWFMPH8DNPMBO EJLEÌSUH FOJOBMBOLBÀDN2EJS 2. f 1 log 481 \" # $ % & 32 p JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS \" 1 # 1 $ 1 8. 33 93 33 3 B % 1 & 1 A \"#$пHFO 933 39 [\"#] m[\"$ ] [ AH ] m [#$] BH = log 3 9 2 3. a3 e2 k ln25 | |H C )$ =MPH364 | |AH =MOY JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS \" # 5 3 5 $ 5 3 25 :VLBSEBLJ WFSJMFSF HÌSF Y BöBôE BLJMFSEFO IBOHJTJEJS % & # F2 $ F3 % F & F6 \" F 9. y 4. log 7 y=logax y=logbx 36 6 JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS \" 1 # 1 $ % & x 49 7 O y=logcx y=logdx 5. eln3 + log 25 + 4log24 õFLJMEF WFSJMFO HSBGJLMFSEFLJ MPHBSJUNB GPOLTJ- ZPOMBSOB HÌSF BöBôEBLJ TSBMBNBMBSEBO IBO- 5 HJTJEPôSVEVS JöMFNJOJOTPOVDVLBÀUS \" B<C < c <E # C< a < c <E \" # $ % & $ D<E< a <C % E< c < a <C & D<E<C< a 1. A 2. E 3. B 4. D 5. B 21 6. D A 8. E 9. C
TEST - 10 -PHBSJUNB'POLTJZPOVOVO(SBGJôJ 1. f^ x h = log3^ x - 2 h 4. f^xh = log 1 ^x + 1h 5 GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ PMBCJMJS PMBCJMJS A) y B) y A) y B) y 2 1 –2 –1 1 x x x x –2 1 –1 y C) y D) y –1 y D) C) 1 1 x 23 4 x –1 x –1 x 13 E) y E) y 1 –1 x 23 5 x 2. y 1 5. 5BONMPMEVôVBSBMLMBSEB 4 x * G Y = log 3 (2x + 1) 1 2 –1 2 x+ 4 m ** G Y = c 5 :VLBSEBHSBGJôJWFSJMFOGPOLTJZPOBöBôEBLJMFS- *** G Y =MPH2 Y- EFOIBOHJTJPMBCJMJS \" G Y =MPH2Y # f^xh = log x GPOLTJZPOMBSOEBOIBOHJMFSJB[BMBOES 2 $ G Y =MPH4Y % f^xh = log x \" :BMO[* # :BMO[** $ :BMO[*** 22 % *WF** & **WF*** & f^xh = log 1 x 2 6. 0 < a <PMNBLÑ[FSF 3. Y=MPH Z=MPH4WF[=MPH3240 Y=MPHa Z=MPHac 29 m , [=MPHa6 5 FöJUMJLMFSJOFHÌSFY ZWF[OJOEPôSVTSBMBOö BöBôEBLJMFSEFOIBOHJTJEJS PMEVôVOBHÌSF Y Z [BSBTOEBLJEPôSVTSBla- NBBöBôEBLJMFSEFOIBOHJTJEJS \" Z<Y<[ # Z<[<Y $ Y< y <[ \" Y< y <[ # Z<[<Y $ Z<Y<[ % Y<[<Z & [< y <Y % [< y <Y & [<Y< y 1. E 2. C 3. A 22 4. D 5. B 6. D
www.aydinyayinlari.com.tr ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 3. MODÜL ·/÷7&34÷5&:&)\";*3-*, ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3*7 Üstel Denklemler ÖRNEK 5 ÖRNEK 1 4x - 5 . 2x - 6 = 0 53x – 4 = 1 125 denkMFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ EFOLMFNJOJTBôMBZBOYEFôFSJLBÀUS 2Y =UPMTVOU2 - 5t - 6 = 0 j (t - 6) (t + 1) = 0 5Y–4 = 5–3 jY- 4 = -3 jY= 1 j x = 1 t - 6 =WFZBU+ 1= 0 3 2Y =WFZBY = -1 Y= log2WFZBq ÖRNEK 2 ÖRNEK 6 12x-1 = 3x + 2 e2x - 8ex + 5 = 0 EFOLMFNJOJTBôMBZBOYEFôFSJLBÀUS EFOLMFNJOJOLÌLMFSUPQMBNLBÀUS xx 12 x 12 = 3 .9 & = 9.12 12 x 3 jd 12 x = 108 & x = 108 & x = log 108 eY =UPMTVO n 4 34 t2 - 8t + 5 = 0 jLÌLMFSU1 ve t2PMTVOU1. t2 = 5 e x 1 x 2 = 5 & ex1 + x2 = 5 .e ÖRNEK 3 Y1 +Y2 = ln5 1 =7 4x – 2 EFOLMFNJOJTBôMBZBOYEFôFSJLBÀUS 4-Y+ 2= 7 j -Y+ 2 = log47 jY= 2 - log47 jY= log416 - log47 j x = log d 16 n 7 4 ÖRNEK 4 ÖRNEK 7 e2x - 5.ex + 6 = 0 f ( x ) = 3x-4 - 2 ve g ( x ) = 2x + 1 - 10 EFOLMFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ PMEVôVOBHÌSF GPH Y =EFOLMFNJOJTBôMBZBOY EFôFSJLBÀUS f^ g^ x h h = g (x) –4 - 2 = 7 3 eY =UPMTVOU2 - 5t + 6 = 0 j (t - 3) (t - 2) = 0 3 g (x) 4 = 9 & g (x) = 6 & g^ x h = 6 3 3 t - 3 =WFZBU-2 = 0 3 eY - 3=WFZBFY - 2 = 0 eY =WFZBFY = 2 H Y = 2Y+1 - 10 = 6 j 2Y.2 = 16 j 2Y = 8 jY= 3 Y=MOWFZBY= ln2 j {ln2, ln3} 1 2. log 108 3. log d 16 n 4. {ln2, ln3} 23 5. {log26} 6. ln5 7. 3 1. 47 4 3
·/÷7&34÷5&:&)\";*3-*, 3. MODÜL ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 www.aydinyayinlari.com.tr -PHBSJUNBM%FOLMFNMFS ÖRNEK 11 %m/*m MPH2 Y+ +MPH2 Y+ = EFOLMFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ a `3+ \\ {}WFC`3PMNBLÐ[FSF MPHaG Y =CJTF MPH2(x2 + 5x + 6) = 1, x + 3 >WFY+ 2 > 0 x2 + 5x + 6 = 2 , x > -WFY> -2 G Y = aCWFG Y >ES x2 + 5x + 4 = 0 MPHaG Y =MPHaH Y JTF G Y =H Y G Y >WFH Y >ES (x + 1) (x + 4) = 0 MPHH Y G Y =MPgH Y I Y JTF x = -WFZBY= -4 G Y =I Y G Y > I Y > H Y >WF x = -EFOLMFNJUBONT[ZBQBS H Y áEJS {-1} ÖRNEK 8 ÖRNEK 12 +MPH-MPH=MPHY log x + log 1 x + log x=4 PMEVôVOBHÌSF YLBÀUS 8 2 2 MPH+MPH-MPH=MPHYWFY>PMNBM logd 10.5 n = log x j 25 = x EFOLMFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ 42 1 log x - log x + 2 log x = 4 WFY> 0 32 2 2 d 1 - 1 + 2 nlog x = 4 jMPH2x = 3 jx=8 j {8} 3 2 ÖRNEK 9 ÖRNEK 13 MPH MPH2 =MPHY log (2x) PMEVôVOBHÌSF YLBÀUS =2 MPH254 =MPH5YWFY>PMNBM log x MPH52 =MPH5x j x = 2 EFOLMFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ ÖRNEK 10 MPHx2x = 2 , x >WFYâ MPH2 MPH +MPH Y- = x2 = 2x j x2- 2x = 0 j x(x - 2) = 0 PMEVôVOBHÌSF YLBÀUS j x =WFZBY= 2, x =EFOLMFNJUBONT[ZBQBS MPH(3 +MPH Y- 1) ) =WFY- 1 > 0 jx > 1 MPH3 (3 +MPH Y- 1 ) ) = 2 {2} MPH3 (3 +MPH Y- 1) ) = 1 3 +MPH Y- 1) = 3 jMPH Y- 1) = 0 j x - 1 = 1 j x = 2 ÖRNEK 14 log x ln x log 4 25 5 - e - 4 4 = 0 EFOLMFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ log 25 ln e log 4 x 5 -x -^ 4h 4 = 0 WFY> 0 x2 - x - 2 = 0 j (x - 2) (x + 1) = 0 j x = 2, x = -1 x = -EFOLMFNJUBONT[ZBQBS {2} 25 9. 2 10. 2 24 11. {–1} 12. {8} 13. {2} 14. {2} 8. 2
www.aydinyayinlari.com.tr ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 3. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 15 ÖRNEK 18 MPHY 2 -MPHY2 - 3 = 0 xlog2^x – 1h = ^x - 1hlog2x EFOLMFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ EFOLMFNJOJOLÌLMFSÀBSQNLBÀUS ^1 4x 4-414hl4og424x2= ^4x4-414h4lo4g42x3 , x > 0 ve x - 1> 0 MPHY 2 -MPHY- 3 =WFY> 0 %FOLMFNJUÑNSFFMTBZMBSJÀJOTBôMBOSWFY> 1 MPHY=UPMTVO t2 - 2t - 3 = 0 j (t - 3) (t + 1) = 0 (1, Þ) t =WFZBU= -1 1 MPHY=WFZBMPHY= -1 j x =WFZB x = 1 10 1000· = 100 10 ÖRNEK 16 ÖRNEK 19 MPHY3 -MPH3Y= 2 ^ 3 + 2 hx + ^ 3 - 2 hx = 4 EFOLMFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ 8 EFOLMFNJOJOLÌLMFSUPQMBNLBÀUS - log x = 2 WFY> 0 log x 3 1 x 3 f 3- 2 hx = 4 p +^ MPH3x =UPMTVO 3- 2 8 - t = 2 j t2 + 2t - 8 = 0 j (t + 4) (t - 2) = 0 ^ 3 - 2 hY = t PMTVO t t = -WFZBU= 2 1 +t=4 j t2 - 4t + 1= ,ÌLMFSU1WFU2PMTVO t MPH3x = - MPH3x = 2 t1.t2 = 1 j ^ 3 - 2 hx1.^ 3 - 2 hx2 = 1 x = 1 WFZB x = 9 & ( 1 , 9 2 ^ 3- 2 h x1+x2 = 1j x1 + x2 = log 1 81 81 ^ 3– 2 h x + x = 0 1 2 ÖRNEK 17 ÖRNEK 20 xlog3x = 9x MPH2 Y2 +MOZ= MPH2 Y +MOZ2 = EFOLMFNTJTUFNJOEFYZLBÀUS EFOLMFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ -MPH2x +MOZ= 3 log x MPH2x +MOZ= MPH2 x = 1 j x = 2 log x 3 = log 9x MPH22 +MOZ= 3 jMOZ= 1 Z= e 33 YZ= 2e MPH3YMPH3x =MPH39 +MPH3x 25 18. ß 19. 0 20. 2e MPH3x)2 = 2 +MPH3x MPH3x =UPMTVO t2- t - 2 = 0 j (t - 2) (t + 1) = 0 j t =WFZBU= -1 MPH3x = MPH3x = -1 1 j ( 1 ,92 x =WFZBY= 33 15. 100 16. ( 1 , 9 2 ( 1 , 9 2 81 3
·/÷7&34÷5&:&)\";*3-*, 3. MODÜL ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 www.aydinyayinlari.com.tr ÖRNEK 21 ÖRNEK 24 3MPHY - 32 -MPHY = MO Y- +MO Y+ =MO Y2 - EFOLMFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ EFOLMFNJOJOÀÌ[ ÑNLÑNFTJOJCVMVOV[ 3 2 MO[(x - 4) (x + 4)] =MO Y2 - 16) log x - = 8 j 3MPHY=UPMTVO MO Y2 - 16) =MO Y2 - JÀJOÀÌ[ÑNSFFMTBZMBSES x - 4 > 0 j x >WFY+ 4 > 0 jx > -4 3 log x PMNBMES Ç = ß 3 t - 9 - 8 = 0 j t2 - 8t - 9 = 0 j (t - 9) (t + 1) = 0 t 3MPHY =WFZBMPHY= -1 MPHY= 2 q x = 100 , Ç = {100} ÖRNEK 22 ÖRNEK 25 MO Y+ =+MOY Y2 - MOB Y+MOB-= 0 JLJODJEFSFDFEFOEFOLMFNJOÀBLöLJLJLÌLÑPMEVôV- PMEVôVOBHÌSF YEFôFSJOFEJS OBHÌSF BLBÀUS 1 D = 0 j MOB 2 - MOB- 1) = 0 2x + 1 > 0 j x > - WFY>PMNBM MOB 2 -MOB+ 4 = 0 MOB- 2)2 = 0 2 MOB- 2 = 0 jMOB= 2 j a = e2 MO Y+ 1) =MOF+MOY MO Y+ 1) =MO YF j 2x + 1 = xe j xe - 2x = 1 1 x(e - 2) = 1 j x = e-2 ÖRNEK 26 log m = 3 + log 1 n ÖRNEK 23 2 2 log clog ^3x - 11hm = 2 FöJUMJôJOFHÌSF log 1 ^m.nhJGBEFTJOJOEFôFSJLBÀ- 22 2 PMEVôVOBHÌSF YEFôFSJOFEJS US MPH2(3x - 11) = 4 j 3x - 11 = 16 MPH2N= 3 -MPH2OjMPH2N+MPH2O= 3 3x =j x = 3 MPH2 NO = 3 jNO= 23 log 1 ^ m.n h = - 2 log ^ m.n h = - 2. log 3 = - 6 22 2 2 21. {100} 1 26 24. (4, Þ) 25. e2 26. –6 22. 23. 3 e-2
·TUFMWF-PHBSJUNBM%FOLMFNMFS TEST - 11 1. 42x – 1 = 1 5. FY -FY + 2 = 0 128 EFOLMFNJOJOÀÌ[ÑNLÑNFTJ{MOB MOC}PMEVôV- OBHÌSF B +CLBÀUS PMEVôVOBHÌSF YEFôFSJLBÀUS \" 7 # $ 5 % & 9 \" # $ % & 4 2 4 2. 3Y- = 4 6. ex + 2 = 3 PMEVôVOBHÌSF YEFôFSJLBÀUS ex EFOLMFNJOJO LÌLMFSJOJO UPQMBN BöBôEBLJMFS- EFOIBOHJTJEJS \" MPH4 # MPH4 $ \" # $ MO & MPH9 % MPH3 % MOF & MOF 3. Y+ +Y -= 0 MPHY=MPH9y EFOLMFNJOJ TBôMBZBO Y EFôFSMFSJOJO UPQMBN PMEVôVOBHÌSF YJMFZBSBTOEBLJCBôOUBöBô- LBÀUS EBLJMFSEFOIBOHJTJEJS \" - # - $ % & $ Z=Y2 \" y = x # Z=Y % Y= y3 & Y2 = y3 4. 9Y -Y+ += 0 EFOLMFNJOJ TBôMBZBO Y EFôFSMFSJOJO ÀBSQN 8. MPH2 = a LBÀUS PMEVôVOBHÌSF aLBÀUS \" 1 # $ % & \" 3 4 # $ 2 2 3 & 2 3 4 % 2 3 2 1. E 2. E 3. B 4. C 5. D 6. C A 8. A
TEST - 12 ·TUFMWF-PHBSJUNBM%FOLMFNMFS 1. MPHY+MPH Y- =MPH 5. MPH2Y+MPH2y =WFlog c x + y m = 1 3 2 EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO PMEVôVOB HÌSF log f x2 + y2 p JGBEFTJOJO EF- IBOHJTJEJS 25 \" \\-^ # \\^ $ \\- ^ ôFSJLBÀUS % \\ ^ & q \" # $ % & 2. MPH3 +MPH2Y = 2 6. MPH Y- -MPH Y+ = PMEVôVOBHÌSF YLBÀUS EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS \" 1 # $ % & \" * - 4 4 # \\-^ $ \\^ 2 3 % * 9 4 & 2 3. MPH4 Y+Z =MPH4Y+MPH4y + PMEVôVOBHÌSF YJOZUÑSÑOEFOFöJUJBöBôEBLJ- MPHY+MPH Y+ =MPH MFSEFOIBOHJTJEJS 3y 3y - 4 3y FöJUMJôJOJTBôMBZBOYEFô FSJLBÀUS \" # $ \" # $ % & 4y - 1 y+1 y-1 3y - 2 3y - 2 % & y+1 y-1 8. MPH2 Y- +MPH2Y= EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS log x - log 4 \" # \\^ $ * - 2 , 1 4 3 4. = ln 9 % * 1, - 1 4 & * 1 , - 2 4 log e 3 3 PMEVôVOBHÌSF YEFôFSJLBÀUS \" # $ % & 1. B 2. E 3. A 4. E 28 5. B 6. E B 8. B
·TUFMWF-PHBSJUNBM%FOLMFNMFS TEST - 13 1. MPH MPH2 MPH3Y = 0 5. MPHY 2 +MPHY- 2 = 0 PMEVôVOBHÌSF YLBÀUS EFOLMFNJOJOLÌLMFSÀBSQNLBÀUS \" # $ % & \" # $ % 1 & 1 10 100 2. log25 x + log625 x = 3 6. MPH2 Y 2 -MPH2Y4 + 3 = 0 2 EFOLMFNJOJOLÌLMFSUPQMBNLBÀUS PMEVôVOBHÌSF YLBÀUS & \" # $ % \" # $ % & 3. MPH4 Y- -=MPH4 Y- EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO MPH9 Y- MPHY3 = IBOHJTJEJS \" * 17 4 # * 23 4 $ * 19 4 EFOLMFNJOJTBôMBZBOYEFô FSJLBÀUS 3 3 2 \" # $ % & % * 39 4 & * 45 4 4 4 4. MPH Y+ +MPH Y+ =MPH+ 8. MPH2 Y - =+Y EFOLMFNJOJOLÌLMFSUPQMBNLBÀUS EFOLMFNJOJOLÌLMFSUPQMBNLBÀUS \" - # $ % & \" - # - $ % & 1. C 2. A 3. A 4. D 29 5. D 6. C B 8. B
TEST - 14 ·TUFMWF-PHBSJUNBM%FOLMFNMFS 1. 9 log 5 + 5 log 9 = 162 5. Y Z` 3+ WFlogf x p = a , log^ x.y h = b x x y FöJUMJôJOJ TBôMBZBO Y EFôFSMFSJOJO ÀBSQN LBÀ- PMEVôVOBHÌSF MPHZBöBô EBLJMFSEFOIBOHJTJOF US FöJUUJS \" - # - $ % & \" a - b # b - a $ B-C 2 2 % C- a & C- 2a 2. Y Z` R+PMNBLÑ[FSF MOY-MOZ+MPH=MO[+ e log 1 3 PMEVôVOBHÌSF YJOZWF[UÑSÑOEFOFöJUJBöBô- 6. x log x = 81 EBLJMFSEFOIBOHJTJEJS 3 EFOLMFNJOJOLÌLMFSÀBSQNLBÀUS \" y z # 3 y2.z $ 3 y2 \" 1 # 1 $ % & z 81 9 z - 2y z + 2y % & 3 3 3. ln^ 9x2 - 4x h = 2 Y- MPH Y- = ^ x - 1 h2 ln^ 4 - 3x h 10 EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS FöJUMJôJOJTBôMBZBOYEFôFSJLBÀUS \" 1 # 3 $ 4 % 5 & 8 \" \\^ # * 24 4 $ * 9 4 2 4 5 6 9 7 2 % * 24 , 11 4 & * 9 , 11 4 7 2 51– log 3x 1 8. log 3 x - log x = 6 5 4. = 12 FöJUMJôJOFHÌSF YEFôFSJLBÀUS PMEVôVOBHÌSF YLBÀUS \" # 24 $ \" # $ % & % 32 & 36 1. D 2. B 3. C 4. D 30 5. B 6. C A 8. E
www.aydinyayinlari.com.tr ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 3. MODÜL ·/÷7&34÷5&:&)\";*3-*, ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3*7 ·TUFMWF-PHBSJUNBM&öJUTJ[MJLMFS ÖRNEK 3 %m/*m x2 - 7x - 18 $ 0 aG Y # aH Y ÐTUFMFõJUTJ[MJLUF 2x–3 - 8 a >JTFG Y #H Y FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ 0 < a <JTFG Y $H Y LPõVMMBSTBóMBONB- x2 -Y- 18 = 0 2x - 3 - 8 = 0 MES (x - 9) (x + 2) = 0 2x - 3 = 8 MPHaG Y #MPHaH Y MPHBSJUNBMFõJUTJ[MJLUF MPHB- x =WFZBY= -2 SJUNBUBONOEBOG Y >WFH Y >TBóMBOS- x=6 LFO x mß –2 6 9 ß a >JTFG Y #H Y + x2mæYm + – – 0 < a <JTFG Y $H Y LPõVMMBSTBóMBONB- MES 2x–3mæ – – + + x2mæYm – + – + 2x–3mæ [-2, 6) b [9, Þ) ÖRNEK 1 ÖRNEK 4 f 5 x–4 25 3x – 2 MPH9 Y+ < FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ p $f p 9 81 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ 5 x– 4 5 6x – 4 5 x + 5 >WFMPH9(x + 5) <MPH99 0 < < 1PMEVôVOEBO d n $d n x > -5 9 >PMEVôVOEBO 99 9 x - 4 # 6x - 4 x+5<9jx<4 0 # 5x j x $ 0 (-5, 4) [ ß ÖRNEK 2 ÖRNEK 5 Y -Y+ +# 0 log c 2x - 4 m # 0 5 x+3 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ 2x - 4 > 0 WFlog d 2x - 4 n # log 1 (5 > 1) x+3 5 x+3 5 2x - 4 #1 x+3 5x =\"PMTVO\"2- 15A + 50 # 0 x-7 #0 (A - 10) (A - 5) = 0 x=2 , x = -3, x+3 5x = 10 5x = 5 x = Y= -3 x mß –3 2 ß + x =MPH510 x=1 2x – 4 + –+ x+3 + 1 log510 + –– + –+ Ymæ x+3 ] 1. [0, Þ) 2. MPH510) 31 3. [–2, 6) b [9, Þ) 4. (–5, 4) 5. >
·/÷7&34÷5&:&)\";*3-*, 3. MODÜL ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 www.aydinyayinlari.com.tr ÖRNEK 6 ÖRNEK 9 log 1 ^4x - 1h $ log 1 ^3x + 2h MPHY+MPH Y+ > FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFT JOFEJS 33 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ x >WFY+ 4 >WFMPH5(x2 + 4x) >MPH5 5 (5 > 1) x2 + 4x > 5 4x - 1 >WFY+ 2 >WFlog 1 ^ 4x - 1 h $ log 1 ^ 3x + 2 h x2 + 4x - 5 > 0 33 1 ve x > - 2 WFd 0 < 1 < 1oldu€undan n x = -5 , x = 1 x> 43 3 4x - 1 # 3x + 2 j x # 3 –5 1 + –+ d 1 ,3G (1, Þ) 4 ÖRNEK 7 ÖRNEK 10 log 1 ^2x - 4h ≤ - 2 MPH2 MPH3 Y+ # FöJUTJ[MJôJOJTBôMBZBOYUBNTBZMBSLBÀUBOFEJS 3 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ 2x - 4 >WFlog 1 ^ 2x - 4 h # log 1 9 d0< 1 <1n x + 1 >WFMPH3(x + 1) >WFMPH2 MPH3(x + 1)) # 1 3 x > -WFMPH3(x + 1) >MPH3WFMPH2 MPH3(x + 1) #MPH22 33 (3 >PMEVôVOEBO >PMEVôVOEBO 2x >WFY- 4 $ 9 x + 1 > MPH3(x + 1) # 2 13 MPH3(x + 1) #MPH39 x > WF x $ x>0 (3 >PMEVôVOEBO 2 13 x+1#9 = ,3n x#8 2 (0, 8] { } ZUBOF ÖRNEK 8 ÖRNEK 11 3 #MPH2 Y- < 4 f^ x h = log^ x + 1 h FöJUTJ[MJLTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJOJCVMVOV[ x - 2 >WFMPH28 #MPH2(x - 2) <MPH216 (2 > 1) x + 1 >WFMPH Y+ 1) $ 0 x >WF# x - 2 < 16 10 # x < 18 x > - MPH Y+ 1) $MPH >PMEVôVOEBO [10, 18) x+1$1 jx$0 [0, Þ) 6. d 1 , 3 G 13 8. [10, 18) 32 9. (1, Þ) 10. 8 11. [0, Þ) 4 = , 3 n 2
www.aydinyayinlari.com.tr ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 3. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 12 ÖRNEK 14 | | MPH4 Y- 3 +MPH42 # 2 MPH TBZTOOUBNLTNY FöJUTJ[MJôJOJTBôMBZBOLBÀGBSLMUBNTBZEFôFSJWBS- \"TBZTOOUBNLTNOOCBTBNBLTBZTZPMNBLÐ[FSF ES MPH\" UJS MPH4 ( 2.| x - 3 | ) #WF| x - 3 | > 0 2 | x - 3 | # WFYâ #VOBHÌSF Y+ZUPQMBNLBÀUS |x - 3| # 8 MPH Y PMTVO -8 # x - 3 # 8 MPH2 <MPH <MPH3 -5 # x # 11 2 < x , ... < 3 [-5, 11] \\ {3} jUBOF x=2 \" TBZTOO UBN LTNOO CBTBNBL TBZT MPH\" TBZTOO UBNLTNOEBOGB[MBPMBDBôOEBO Z= 13 x +Z= 15 ÖRNEK 13 ÖRNEK 15 log7^ x + 12 h # 2 MPH= log7x PMEVôVOBHÌSF 25TBZTLBÀCBTBNBLMES FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ MPH25 =MPH= MPH+MPH = 25.(1 +MPH = 25.(1 + 0,90309) = MPHx(x + 12) # 2 x >JTFY2 $ x + 12 j x2 - x - 12 $ 0 + 1=CBTBNBLMES –3 4 (x - 4) (x + 3) = 0 Üstel ve Logaritmik Fonksiyonlarla İlgili Gerçek + –+ x = 4, x = -3 Hayat Problemleri ÖRNEK 16 x >PMEVôVOEBO[4, Þ) 3BEZPBLUJGCJSNBEEFOJOCJSJN[BNBOEBCP[VMNBPMBT- 0 < x<JTFY2 # x + 12 j x2 - x - 12 # 0 MLTBCJUJmPMNBLÐ[FSF CVNBEEFOJOZBSMBONBTÐSFTJ t = ln 2 GPSNÐMÐJMFIFTBQMBOS –3 4 x = 4, x = -3 + –+ 0 < x <PMEVôVOEBO m 3BEZPBLUJGCJSNBEEFOJOCJSJN[BNBOEBCP[VMNBPMBT- Ç = (0, 1) b [4, Þ) ML TBCJUJ PMEVóVOB HËSF CV NBEEFOJO NFWDVU NJLUBSOO 1 TJLBEBSB[BMNBTJÀJOZBLMBöLLBÀZM %m/*m 7 O`;PMTVOMPHO =OMPH=O HFÀNFTJHFSFLJS MO VO UBN TBZ LVWWFUJ õFLMJOEF ZB[MBNBZBO t = ln 2 = 0, 693 = 77 ZM QP[JUJG CJS HFS¿FL TBZOO POMVL MPHBSJUNBT BS- 0, 009 0, 009 EõLJLJUBNTBZBSBTOEBES 11 EFO CÐZÐL CJS HFS¿FL TBZOO POMVL MPHBSJU- TJLBEBSB[BMNBTNJLUBSOO JPMNBTEFNFLUJS NBTQP[JUJGUJS 78 Y EFOCÐZÐLCJSHFS¿FLTBZPMTVOYTBZT- #VEVSVNLFSFZBSMBONBTEFNFLUJS OOMPHBSJUNBTOOUBNLTNBJTFB+ YTB- =ZM ZTOOUBNLTNOOCBTBNBLTBZTOWFSJS 12. 16 13. (0, 1) b< ß 33 14. 15 15. 48 16. 231
·/÷7&34÷5&:&)\";*3-*, 3. MODÜL ·45&-7&-0(\"3÷5.÷,'0/,4÷:0/-\"3 www.aydinyayinlari.com.tr ÖRNEK 17 ÖRNEK 19 3JDIUFS ËM¿FóJ TJTNPMPKJEF LVMMBOMBO EÐOZB HFOFMJOEF #JS CËMHFEF CBõMBOH¿UBLJ JOTBO TBZT J U ZM TPOSBLJ NFZEBOBHFMFOEFQSFNMFSJOBMFUTFMCÐZÐLMÐLMFSJJMFTBS- JOTBOTBZTJ2 CËMHFEFLJJOTBOBSUõI[LPMNBLÐ[FSF TOUPSBOOCFMJSMFZFOWFTOGMBSBBZSBOVMVTMBSBSBTËM- CËMHFEFLJJOTBOTBZT ¿ÐCJSJNJEJS.JLSPODJOTJOEFOËM¿ÐMFONBLTJNVNHFOMJL J2 =J +L U2 -U EWFPMVõBOTBSTOUOO3JDIUFSËM¿FóJOFHËSFCÐZÐLMÐóÐ CJ¿JNJOEFNPEFMMFONJõUJS 3PMNBLÐ[FSF EFQSFNJOõJEEFUJ #JSNBIBMMFEFZMOEBZBQMBOTBZNEBLJ- 3=MPHE öJ ZMOEB ZBQMBO TBZNEB LJöJ PMEVôV- OBHÌSF JLJZMEBNBIBMMFEFLJBSUöI[ZÑ[EFLBÀUS CJ¿JNJOEFIFTBQMBOS NN=3NJLSPOWFMPH PMNBLÐ[FSF NBLTJ- 12852 = 8925 . ( 1 +L 2 NVNHFOMJôJNNPMBSBLÌMÀÑMFOEFQSFNJO3JDI- 1,44 = ( 1 +L 2 UFSÌMÀFôJOFHÌSFCÑZÑLMÑôÑLBÀUS 12 NN= 490.103NJLSPO= 49.104NJLSPO =1+k R=MPH 4) 10 =MPH2 +MPH4 2 = 2.0,85 +MPH k = = 0, 2 = + 4 = 10 %20 ÖRNEK 18 ÖRNEK 20 ,BSCPO-JMFZBõUBZJOJ ZBLMBõLZMBLBEBSCJ- #JMFõJL GBJ[ CBOLBZB ZBUSMBO QBSBOO EËOFN TPOVOEB ZPMPKJL PSJKJOMJ BSLFPMPKJL OFTOFMFSJO ZBõO CFMJSMFNFEF BOBQBSBWFGBJ[CJSMFõUJSJMFSFLFMEFFEJMFOUVUBSOÐ[FSJO- LVMMBOMBO CJS ZËOUFNEJS :Bõ UBZJOJ ZBQMBDBL OFTOFOJO ZBõYPMNBLÐ[FSF ZLBSCPO-NJLUBSOOUÐNLBSCPO EFOUFLSBSGBJ[BMOBSBLWFIFSEËOFNUFLSBSMBOBSBLFM- NJLUBSOBPSBOOHËTUFSJSLFO OFTOFOJOZBõ EFFEJMFOCJSGBJ[¿FõJUJEJS#BOLBZBCJMFõJLGBJ[JMFZBUS- MBOQBSBEBOFMEFFEJMFOGBJ[UVUBSG ZBUSMBOQBSB\" GBJ[ log y ZÐ[EFTJO ZBUSMBOEËOFNTBZTUPMNBLÐ[FSF x = - 5730· f = A.c1 + n t log 2 EFOLMFNJJMFIFTBQMBOS 100 m -A \"SLFPMPKJLCJSLB[EBCVMVOBOJTLFMFUUFOLBEBSLBS- EFOLMFNJJMFIFTBQMBOS CPO-J¿FSEJóJIFTBQMBONõUS #BOLBZBZMMLGBJ[JMFCJMFöJLGBJ[EFZBUSMBO #VOBHÌSF JTLFMFUJOZBLMBöLZBöLBÀUS MJSBOOZMTPOSBHFUJSFDFôJGBJ[LBÀMJSBES MPH 25 3 f = 768.d 1 + n - 768 100 40 f = 768. 125 - 768 log log 4 - log 10 64 100 j x = - 5730. x = - 5730· f = 1500 - 768 log 2 log 2 2 log 2 - 1 0, 6 - 1 f = 732 x = - 5730· log 2 j x = - 5730· 0, 3 0, 4 4 x = 5730· = 5730· = 0, 3 3 18. 34 19. %20 20.
-PHBSJUNBM&öJUTJ[MJLMFS TEST - 15 1. MPH2 Y+ < 5. x `[ PMNBLÑ[FSF FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJOEF LBÀ UBN TBZ MPH2 TJOY > - WBSES FöJUTJ[MJôJOJO ÀÌ[ÑN BSBMô BöBôEBLJMFSEFO \" # $ % & IBOHJTJEJS \" # $ % & 2. MPH Y+ 2 >MPH 6. G Y =MPH MPH3 MPH2Y FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJBöBôEB- IBOHJTJEJS LJMFSEFOIBOHJTJEJS \" Þ # Þ $ Þ % Þ & Þ \" Þ # Þ $ Þ % Þ & Þ 3. log 1 ^x - 2h 2 log 1 ^7 - xh f^ x h = e2x + x3 + log3^ x + 1 h 33 GPOLTJZPOVOVOFOHFOJöUBONBSBMôBöBôEB- LJMFSEFOIBOHJTJEJS FöJUTJ[MJôJOJO ÀÌ[ÑN LÑN FTJOEF LBÀ UBN TBZ EFôFSJWBSES \" - Þ # - > $ <- > \" # $ % & % Þ & < Þ 4. MPH4 Y-å +MPH4 Y+ # 2 8. f ^xh = ln f log 1 ^x - 3hp FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJOEF LBÀ UBN TBZ 3 WBSES GPOLTJZPOVOVOFOHFOJöUBONBSBMôBöBôEB- LJMFSEFOIBOHJTJEJS \" # $ \" # $ % & % & 1. D 2. D 3. E 4. A 35 5. A 6. C E 8. B
TEST - 16 -PHBSJUNBM&öJUTJ[MJLMFS 1. MPH Y- 2 - 9 # 0 5. MOY 2 -MOY2 # 0 FöJUTJ[MJôJOJTBôMBZBOLBÀGBSLMYUBNTBZEFôF- FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO SJWBSES IBOHJTJEJS \" # $ \" f 1 , eH # > 1 , eH $ > 1 , 1H e e e % & % < F> <& F2> 2. log x 6. 3 log 5 + 5 log 3 $ 50 x x x 7 < 2401 FöJUTJ[MJôJOJTBôMBZBOFOCÑZÑLWFFOLÑÀÑLUBN FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO TBZMBSOUPQMBNLBÀUS IBOHJTJEJS \" 3+ # 3+=\\^ $ 3 > \" # $ % & % ^ 0, 3 @ \\ \" 1 , & q 3. lnclog c x + 2 mm # 0 -3 # log 1 x < 43 2 FöJUTJ[MJôJOJTBôMBZBOLBÀYUBNTBZEFôFSJWBS- FöJUTJ[MJLTJTUFNJOJTBôMBZBOLBÀGBSLMYUBNTB- ES ZEFôFSJWBSES \" # $ % & \" # $ % & 4. log 1 ^x2 - 8x + 13h $ 0 8. MPH Y+ +MPH Y- #MPH Y+ 3 FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS FöJUTJ[MJôJOJTBôMBZBOGBSLMYUBNTBZEFôFSMFSJ UPQMBNLBÀUS \" <- > # - $ <- > \" # $ % & % > & - 1. C 2. D 3. B 4. A 36 5. D 6. C E 8. D
-PHBSJUNBM&öJUTJ[MJLMFS(FSÀFL)BZBU1SPCMFNMFSJ TEST - 17 1. 3 #MPH2 Y- < 4. MPH= WFMPH= FöJUTJ[MJôJOJTBôMBZBOLBÀGBSLMUBNTBZEFôFSJ PMEVôVOBHÌSF 50TBZTLBÀCBTBNBLMES WBSES \" # $ % & \" # $ % & 5. ^ 5x - 2 h.^ 2x - 7 h.^ 4 - 25x h # 0 2. f^xh = 2 log 1 ^x - 4h + 3 PMNBLÑ[FSF FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS 3 \" 9log 2 , 3k # 9log 2 , log27C f ( x ) + 1 $öBSUOTBôMBZBOYUBNTBZEFôFS- MFSJOJOUPQMBNLBÀUS 5 5 \" # $ % & $ a- 3 , log52C % 9log52, 4k & 9log27, 3 k , %log 2 / 5 3. YLJõJOJOCVMVOEVóVCJSÐMLFJ¿JOLOÐGVTBSUõPSBO 6. #JS CBOLBZB ZBUSMBO QBSBOO CJMFõJL GBJ[ PSBO O PMNBLÐ[FSF UZMTPOSBLJOÐGVT ZBUSMBOQBSB\"MJSBPMNBLÐ[FSF UZMTPOSBCJSJLFO y =YFLU QBSBNJLUBS CJ¿JNJOEFNPEFMMFONJõUJS y =\"FOU :MMLOÑGVTBSUöPSBOOOPMEVôVCJSÑMLF- CJ¿JNJOEFNPEFMMFOJS OJOOÑGVTVZBLMBöLLBÀZMTPOSBEÌSULBUBSUBS (ln5 1,60) #BOLBZBZBUSMBOQBSBOO 7 LBULBEBSBSUNB- 2 \" # $ % & TJÀJOCJMFöJLGBJ[JMFLBÀZMMôOBCBOLBZB ZBUSMNBMES (ln3,5 1,5) \" # $ % & 1. C 2. E 3. A 4. D 5. E 6. B
TEST - 18 -PHBSJUNBM&öJUTJ[MJLMFS(FSÀFL)BZBU1SPCMFNMFSJ 1. YMOY #F2Y 3. Y -Y+ +# 0 FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS IBOHJTJEJS \" > 1 , e2H # f - 3 , 1 H , \" e2 , \" 9log 10 , 3k # 9log52 , 3C e e 5 $ 9e2 , 3 k , * 1 4 % > 1 ,3p $ [ ß % 91, log510C e e2 & 9log52 , log510C & > 1 , eH e2 2. MJUSF ¿Ë[FMUJEF ¿Ë[ÐONÐõ NPM TBZT NPMBSJUF PM- 4. ;FNJOEF PMVõBO TBSTOUOO CÐZÐLMÐóÐ 3 PMNBL NBLÐ[FSF ¿Ë[FMUJOJOQ)EFóFSJ Ð[FSF Q)= -MPH[H+] R = logf I p I0 GPSNÐMÐJMFIFTBQMBOS GPSNÐMÐJMFIFTBQMBOS &óFS I TBSTOUõJEFUJ I0ËM¿ÐMFCJMFONJOJNVNTBSTOUõJE- EFUJEJS r Q)=JTF¿Ë[FMUJOËUSBM r Q)>JTF¿Ë[FMUJCB[JL 3JDIUFSÌMÀFôJOFHÌSF CÑZÑLMÑôÑ PMBSBLÌM- ÀÑMFO EFQSFN TBSTOUTOO JML BSUÀ TBSTOUT r Q)<JTF¿Ë[FMUJBTJEJLUJS EFQSFNJO 1 LBU öJEEFUMJ PMEVôVOB HÌSF #VOBHÌSF 1000 * [H+] EFSJõJNJ 10–11 NPMBSJUF JTF ¿Ë[FMUJ BTJ- BSUÀ öJEEFUJO CÑZÑLMÑôÑ LBÀ PMBSBL ÌMÀÑMNÑö- EJLUJS UÑS ** Q) EFóFSJ PMBO ¿Ë[FMUJEF [H+] EFSJõJNJ \" # $ % & –78 10 25 NPMBSJUFEJS *** /ËUSBM¿Ë[FMUJEF[H+]EFSJõJNJ-3UÐS JGBEFMFSJOEFOIBOHJMFSJEPôSVEVS \" :BMO[* # :BMO[** $ :BMO[*** % *WF** & *WF*** 1. A 2. D 38 3. D 4. C
·TUFMWF-PHBSJUNJL'POLTJZPOMBS KARMA TEST - 1 1. MPHa= 4 5. f^xh = log 9log3 ^3x - 2hC 2 PMEVôVOBHÌSF MPH8BLBÀUS PMEVôVOBHÌSF G-1 LBÀUS \" # 3 $ 1 % 1 & 1 \" # 83 $ 67 % & 11 2 4 3 2 33 3 6. log 16 . log 125 . log 49 = log 4 x ^8xh 2. MPH2Y= 57 8 PMEVôVOBHÌSF 17 x50 JGBEFTJOJOFöJUJLBÀUS PMEVôVOBHÌSF YLBÀUS \" # $ % & \" # $ % & 3. f_ x i = log2_ x + 3 i MPH=Y MPH=Z MPH=[ LVSBM JMF WFSJMFO G GPOLTJZPOVOVO FO HFOJö UB- PMEVôVOBHÌSF MPHOJOY Z [UÑSÑOEFOFöJUJ ONLÑNFTJBöBôEBLJMFSEFOIBOHJTJEJS BöBôEBLJMFSEFOIBOHJTJEJS \" -3, - # -2, R $ -R, - \" Y+ y +[ # Y- y +[ $ Y+ 2y +[ % -3, -> & <-2, R % Y- 2y +[ & Y- 3y +[ 4. f_ x i = 23x + 1 1 1 1 1 PMEVôVOBHÌSF G-1 Y BöBôEBLJMFSEFOIBOHJTJ- 8. + logab logba 1- 1- EJS \" log2x + 1 # log3x + 1 $ log3x - 1 JöMFNJOJOTPOVDVBöBôEBLJMFSEFOIBOHJTJEJS 2 22 \" MPHaC-MPHCB # MPHa BC % a % log2x + 1 & log2x - 1 $ MPHa f b p 33 a b & 1. D 2. B 3. E 4. E 39 5. B 6. A C 8. E
KARMA TEST - 2 ·TUFMWF-PHBSJUNJL'POLTJZPOMBS 1. log 9log ^ ln e512 hC 5. 4 + 9 + log63 log 3 6 log3 2 6 32 JGBEFTJOJOFöJUJLBÀUS \" # $ % & JöMFNJOJOTPOVDVLBÀUS \" # $ % & 2. f^ x h = log ^ 7x - x2 h ^x–3h GPOLTJZPOVOV UBONM ZBQBO Y UBN TBZMBSOO 6. aY =Cy UPQMBNLBÀUS \" # $ % & PMEVôVOB HÌSF logabb JGBEFTJOJO FöJUJ BöBô- EBLJMFSEFOIBOHJTJEJS x.y x-y x+y \" # $ x-y x+y xy 3. f: R Z3PMNBLÑ[FSF xy & x % x+y G Y =FY+F x+y PMEVôVOB HÌSF G-1 Y JO LVSBM BöBôEBLJMFS- EFOIBOHJTJEJS \" FYmF # FYm $ MO YmF % MOYmF & ln x Y2 +Y-MPH4N= 0 e EFOLMFNJOJOÀBLöLJLJHFSÀFMLÌLÑOÑOPMNBT JÀJONOFPMNBMES \" 1 # 1 $ % 3 & 2 4 2 4. A \"#$EJLпHFO log916 B I I#$ =MPH9CS % m ( ACB ) = 15° 15° C & 8. MPH= a :VLBSEBLJWFSJMFSFHÌSF A^ ABC hLBÀCJSJNLB- PMEVôVOBHÌSF MPHOJOBUÑSÑOEFOEFôFSJOF- SFEJS EJS \" 1 log 3 2 # log 3 2 $ 1 a log 2 2 2 2 k 3 \" -B # -B $ - 2a % a log 2k2 & log34 % -B & -B 3 1. B 2. B 3. D 4. C 40 5. C 6. E A 8. D
·TUFMWF-PHBSJUNJL'POLTJZPOMBS KARMA TEST - 3 1. a > 0 5. log 1 ^x - 3h = x -9 2 ax + 6a–x - 5 = 0 3 EFOLMFNJOJOLËLMFSJYWFY2PMTVO EFOLMFNJOJOÀÌ[ÑNLÑNFTJLBÀFMFNBOMES x1 + x2 = 2 \" # $ PMEVôVOBHÌSF BLBÀUS % & TPOTV[ \" 6 # $ 2 3 % 3 2 & 6. \"öBôEBLJ EFOLMFNMFSEFO IBOHJTJOJO HSBGJôJ ZBOMöUS A) y B) y y = log3x 1 y = log2(x–3) 3x O 1 3 45 x O1 2. f_ x i = 4 log_ x2 – 4 i + x - log x C) D) y y y = e–x GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJBöBôEB- 1 x 11 x LJMFSEFOIBOHJTJEJS O y O1 y=log 1x \" _ 0, 2 i , _ 5, 3 i # 7 5, 3 i E) 2 2 $ R % _ – 5, 0 i , _ 5, 3 i & R 1 –2 –1 x 5 O – 3 y = log1(x + 2) 3 3. G Y =MPH3Y HPG Y =Y+ _ log 2 i2 + log 4. log 5 + _ log 5 i2 PMEVôVOB HÌSF H BöBôEBLJMFSEFO IBOHJTJ- EJS \" # $ % & JGBEFTJOJOFöJUJLBÀUS \" # $ % & 4. MPH3 = a 8. MPH= PMEVôVOB HÌSF MPH3 JGBEFTJOJO B DJOTJO- PMEVôVOBHÌSF 50TBZTLBÀCBTBNBLMES EFOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS \" # $ % & \" B- # B- $ B- 3 % B- & B- 1. A 2. B 3. E 4. D 41 5. B 6. C A 8. D
KARMA TEST - 4 ·TUFMWF-PHBSJUNJL'POLTJZPOMBS 1. MPH3N=B MPH3O=CWFMPH9 NO = c 5. Y2 +Y+MPH2 B- = 0 PMEVôVOBHÌSF B C DBSBTOEBLJCBôOUBöBô- EFOLMFNJOJOHFSÀFLTBZMBSEBLJÀÌ[ÑNLÑNFTJ EBLJMFSEFOIBOHJTJEJS CPö LÑNF PMEVôVOB HÌSF B OO BMBDBô FO LÑ- ÀÑLUBNTBZEFôFSJLBÀUS \" B-C- c = # B+C+ c = 0 \" # $ % & $ B+C- 2c = % B+C- c = 0 & a +C-= 0 2. 6ZHVOöBSUMBSEBUBONMG Y WFH Y GPOLTJZPO- 6. 4Y+ - 2Y+ - 20 = 0 MBSJÀJO PMEVôVOBHÌSF YBöBôEBLJMFSEFOIBOHJTJEJS f^ x h = 2x ve g^ x h = log2^ x + 3 h ln 5 - 1 ln 2 - 1 ln 2 - ln 5 PMEVôVOBHÌSF a fog–1 k_ 3 iLBÀUS \" # $ \" # $ % & ln 2 ln 5 ln 5 1 + ln 5 ln 5 - ln 2 % & ln 2 ln 2 3. logf 1– 1 p + logf 1– 1 p +. . . + logf 1– 1 p 23 10000 JGBEFTJOJOEFôFSJBöBôEBLJMFSEFOIBOHJTJEJS a log 2 x k2 - log x > 0 2 \" - # - $ - % - & - FöJUTJ[MJôJOJOFOHFOJöÀÌ[ÑNBSBMôBöBôEBLJ- MFSEFOIBOHJTJEJS \" (0, 2) , (4, 3 ) # (1, 2) , (4, 3 ) $ (0,1) , (2, 3 ) % R 4. y & y = f(x) O 23 x G Y =MPHa CY+D GPOLTJZPOVOVOHSBGJóJõFLJMEFLJHJ- 8. logx 7 - 2 10 + logx_ 3 2 + 3 5 i = 2 CJEJS f-1 ( 2 ) =PMEVôVOBHÌSF BLBÀUS FöJUMJôJOJTBôMBZBOYEFôFSJLBÀUS \" # $ % 1 & 1 2 5 \" # $ % & 1. C 2. D 3. B 4. B 42 5. C 6. E C 8. A
·TUFMWF-PHBSJUNJL'POLTJZPOMBS KARMA TEST - 5 1. \"öBôEBLJMFSEFOIBOHJTJCJSSBTZPOFMTBZES 5. Y +–Y = \" F # MO $ ln e e EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS % ln e 5 & ln3 e2 \" \\^ # \\MPH4^ $ \\MPH4 } % % 0 , log45 / & % 0 , log54 / 2. \"öBôEBLJFöJUTJ[MJLMFSEFOIBOHJTJZBOMöUS \" MPH2> 0 # log 1 1 >0 $ log 1 < 0 6 3 4 16 % log c 1 m > 0 6. log x 32 49 7 = 12 - x & ln c 1 m < 0 PMEVôVOBHÌSF YLBÀUS 5 \" - # $ % & 3. MPH=YWFMPH= y PMEVôVOBHÌSF MPH4JOYWFZUÑSÑOEFOFöJ- UJBöBôEBLJMFSEFOIBOHJTJEJS 2x +y 2x +y 2x +y 1 <BãCJÀJO \" # $ 2y–2x 2–2x 2 +xy & 2 - xy % 2 + x MPHaC+MPHCa 2x +y 2x +y UPQMBNBöBôEBLJMFSEFOIBOHJTJPMBNB[ \" 7 # $ 5 % & 3 2 2 2 y 4. O 2x –3 a –1 8. 2 + log5^ x + 1 h = 1 ôFLJMEFLJHSBGJL y = logb_ x + 3 iGPOLTJZPOVOBBJU- EFOLMFNJOJ TBôMBZBO Y EFôFSMFSJOEFO CJSJ IBO- UJS HJTJEJS #VOBHÌSF BCLBÀUS \" - # - $ - 1 4 \" - # - $ - 4 5 % - 2 % - 1 & - 4 5 & - 1 5 5 5 1. E 2. D 3. B 4. D 43 5. E 6. D E 8. E
KARMA TEST - 6 ·TUFMWF-PHBSJUNJL'POLTJZPOMBS 1. \"öBôEBLJMFSEFOLBÀUBOFTJQP[JUJGUJS 5. e2x + 3ex - 10 = 0 * MPH3 EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS ** log 5 3 1 2 \" \\MO^ # \\MO ^ $ \\MO^ *** log 1 2 % \\MO^ & \\^ 7 3 *7 MO 7 log 2 3 \" # $ % & 6. f^xh = log f x3 + 1 p 5 1- x 2. a =MPH3 C=MPH D=MPH PMEVôVOBHÌSF G Y <PMNBTJÀJOYSFFMTB- PMEVôVOBHÌSF B C DBSBTOEBLJTSBMBNBBöB- ZMBSBöBôEBLJBSBMLMBSEBOIBOHJTJOEFPMNBM- ôEBLJMFSEFOIBOHJTJE JS ES \" Y< # Y> $ -<Y< 0 \" B<C<D # C< a <D $ D<C< a % -<Y< & Y<WFY> % D< a <C & B< c <C 3. MPH MOY +MPH MOY2 - = 0 3 - log ^2x - 4h # 2 EFOLMFNJOJOLÌLMFSÀBSQNLBÀUS 2 \" F # $ F FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOEFLBÀUBOFYUBN TBZEFôFSJWBSES A # $ % & % F F & Fm 4. 10.xlogx = x2 2 – log x 3 8. f^xh = -x2 - x + 2 EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJBöBôEB- LJMFSEFOIBOHJTJEJS \" * 1 , 10 4 # \\^ $ \\ ^ \" # ] 10 $ - {} % ] - {} % * 1 , 100 4 & * 1 , 100 4 & [ - {} 10 100 1. D 2. C 3. C 4. B 44 5. D 6. C B 8. D
·TUFMWF-PHBSJUNJL'POLTJZPOMBS KARMA TEST - 7 1. MPH6 2 +MPH6MPH6+ MPH6 2 5. xlog25 - 25log2x = 0 JöMFNJOJOTPOVDVLBÀUS EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS \" 1 2 # $ % & \" \\^ # \\^ $ \\^ % \\ ^ & \\ ^ 2. log ^a.bh = log 625 25 log c a m = log 49 2b 7 PMEVôVOBHÌSF BOOQP[JUJGEFôFSJLBÀUS 6. log f 1 _ x + y i p = 1 _ log x + log y i \" # $ % & 32 PMEVôVOBHÌSF x + y JGBEFTJOJOFöJUJLBÀUS yx \" # $ % & 3. a >PMNBLÐ[FSF x = log 4 , y = log 9 , z = log 25 log 25 a log ^x - 3hk > –1 aaa 0, 2 PMEVôVOBHÌSF Y Z [BSBTOEBLJTSBMBNBBöB- ôEBLJMFSEFOIBOHJTJEJS FöJUTJ[MJôJOJTBôMBZBOLBÀUBOFYUBNTBZTWBS- \" [>Y>Z # [> y >Y $ Y> y >[ ES % Y>[>Z & Z>[>Y \" # $ % & 4. BCâPMNBLÑ[FSF 8. MPH=Y MPHa.ba =OPMEVôVOBHÌSF PMEVôVOBHÌSF MPH4JOYDJOTJOEFOFöJUJOF- 3a EJS loga.b b \" x + 2 # x $ x + 2 JGBEFTJBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS 2x 2x + 2 2 \" 3n - 5 # 3 + 5n $ 2n - 3 % 2x + 1 & x + 2 3 2 4 x 2x + 1 % 5n - 3 & 3n + 2 6 5 1. D 2. C 3. B 4. D 45 5. B 6. D D 8. A
KARMA TEST - 8 ·TUFMWF-PHBSJUNJL'POLTJZPOMBS 1. y =G Y =MPH3 Y- + 2 x GPOLTJZPOVOVOUFSTGPOLTJZPOVOVOHSBGJôJBöB- 4. y = 5 ôEBLJMFSEFOIBOHJTJPMBCJMJS GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- EJS A) y B) y A) y B) y 3 2 1 1 2 xO x x O2 O O x 3 C) y D) y C) y D) y 4 3 O1 x 1 x 3 2 O x O2 O2 x E) y E) y 1 x O 2 O2 3x 5. log 1 ^16x - 20h < - x 4 FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO 2. MOY+-MOY = 6 IBOHJTJEJS EFOLMFNJOJO LÌLMFS UPQMBN BöBôEBLJMFSEFO \" alog45 , 3k # R IBOHJTJEJS % R $ a log 20 , 3k \" F # F+ $ F 4 & R % F+ & F 3. MPH Y- +MPH Y+ =MPH Y+ 6. MPH4= a PMEVôVOBHÌSF MPH6BöBôEBLJMFSEFOIBOHJ- TJOFFöJUUJS FöJUMJôJOJTBôMBZBOYEFôFSMFSJOJOUPQMBNLBÀ- \" 2 + 3a # 3 - 2a $ 5 + 2a US 3 + 3a 4 + 2a 2 + 3a \" # $ % & - % 6 + 2a & 1 + 3a 3 + 2a 2a 1. A 2. B 3. A 46 4. E 5. A 6. D
Üstel ve Logaritmik Fonksiyonlar <(1m1(6m/6258/$5 1. 5SBGJLUF DF[B JõMFNJ ZBQMBO CJS TÐSÐDÐ ËEFNFZJ 3. 6MVTMBSBSBTSFGFSBOTTFTõJEEFUJI0 =-XBUUN2 IFNFOZBQBSTBHFDJLNFGBJ[JBMONBNBLUBES$F- EJS±M¿ÐMFOTFTõJEEFUJIXBUUN2PMNBLÐ[FSF TFT [BM JõMFNJO $ MJSB PMEVóV CJS ÐDSFUJ U BZ HFDJLUJSFO TÐSÐDÐOÐOËEFZFDFóJGBJ[NJLUBS'MJSBPMNBLÐ[F- EÐ[FZJ SF '=$ U L =MPH I dB I0 JMFNPEFMMFONJõUJS JMFNPEFMMFONJõUJS ,FOEJTJOF CJMEJSJMFO DF[BZ BZ HFDJLUJSFO TÐSÐDÐ GBJ[JJMFCJSMJLUFMJSBËEFNJõUJS #VOBHÌSe, * 4FT õJEEFUJ - waUUN2 PMBO Jõ NBLJOFTJOJO 4ÑSÑDÑ HFMFO DF[BZ HFDJLUJSNFEFO ÌEFTFZEJ LBÀMJSBÌEFSEJ TFTEÐ[FZJE#EJS ** 4FTõJEEFUJ6LBUBSUUSMEóOEBTFTEÐ[FZJ \" # $ % & BSUBS *** 4FTEÐ[FZJE#PMBOTFTJOõJEEFUJ -XBUUN2EJS CJMHJMFSJOEFOIBOHJMFSJEPôSVEVS \" :BMO[* # :BMO[** $ :BMO[*** % *WF*** & * **WF*** 2. 6ZHVOLPõVMMBSOWBSPMEVóVCJSPSUBNEBUTBOJZFEF 4. #JSNBóB[BOOZBQMBOYMJSBMLBMõWFSJõTPOVDVOEB LBCOJ¿JOEFCVMVOBOCBLUFSJOJOBóSMóYNJMJHSBN NBóB[BLBSUOBZÐLMFEJóJ1 Y MJSBPMBOQBSBQVBO PMNBLÐ[FSF 1 Y =MPH2Y t JMFNPEFMMFONJõUJS Y= 2 4 .BóB[BOO LºS JMF MJSBZB TBUUó HËNMFóJ BMBO&SPM ËODFLJMJSBMLBMõWFSJõMFSJOEFOFMEF JMFNPEFMMFONJõUJS FUUJóJQBSBQVBOOËEFNFEFLVMMBONõUS ¶SFZFOCBLUFSJMFSJOUFIMJLFMJTFWJZFZFVMBõNBTJ¿JO .BôB[BOO HÌNMFL TBUöOEBO FMEF FUUJôJ L»S CBLUFSJTBZTOOEFOGB[MBPMNBTHFSFLNFL- LBÀMJSBES UFEJS \" # $ % & #VOBHÌSF LBÀODTBOJZFEFCBLUFSJTBZTUFI- MJLFTFWJZFTJOJOTOSOBVMBöS \" # $ % & 1. C 2. D 3. D 4. A
<(1m1(6m/6258/$5 Üstel ve Logaritmik Fonksiyonlar 1. ,BZBLMBBUMBNBZBQBOCJSTQPSDVOVOSBNQBEBOBZ- 3. 1MBKEB CVMVOBO LVN UBOFDJLMFSJOJO NN DJOTJOEFO SMELUBOTPOSBZBUBZEPóSVMUVEBBMEóIFSYNNF- WFSJMFOPSUBMBNB¿BQE FóJNJNPMNBLÐ[FSF TBGFEF[FNJOFEFóEJóJOPLUBZBEÐõFZEPóSVMUVEBLJ V[BLMóG Y PMNBLÐ[FSF N= MPHE+ G Y =MPH2 -Y CBóOUTWBSES PMBSBLNPEFMMFONJõUJS 1MBKEBOBMOBOLVNUBOFDJLMFSJOJO¿BQMBSOOPSUBMB- y NBT NNPMBSBLIFTBQMBONõUS #VOBHÌSF CVQMBKOFôJNJLBÀUS (log0,06 , -1,2) x \" # $ O zemin % & #VOBHÌSF * :BUBZEPóSVMUVEBNFUSFZPMBMBOTQPSDVEÐ- õFZEPóSVMUVEBNZPMBMNõUS ** ;FNJOEFONFUSFZÐLTFLMJLUFZLFOZBUBZEPó- SVMUVEBNFUSFZPMBMNõUS *** 3BNQBEBO BZSMEó BOEBO [FNJOF UFNBTOB LBEBSEÐõFZEPóSVMUVEBNFUSFZPMBMNõUS JGBEFMFSJOEFOIBOHJMFSJEPôSVEVS \" :BMO[* # :BMO[** $ :BMO[*** % *WF** & **WF*** 2. #JS¿FLJSHFOJOCJSJODJT¿SBZõOEBMPHNFUSF 4. #JSCËMHFEFCVMVOBOJOTBOTBZTÐTTFMZËOUFNJMF JLJODJ T¿SBZõOEB MPH NFUSF пÐODÐ T¿SBZ- IFTBQMBOSLFO 10 JML OÐGVT S ZMML PSUBMBNB OÐGVT õOEBMPHNFUSFPMBDBLõFLJMEFOT¿SBZõO- BSUõ I[ U JML OÐGVT IFTBCOEBO TPOSB HF¿FO [B- EBBMEóZPMG O PMNBLÐ[FSF NBO UZMTPOSBQMBOMBOBOOÐGVT1PMNBLÐ[FSF G O =MPH O+ 4 1=10FSU PMBSBLNPEFMMFONJõUJS CJ¿JNJOEFNPEFMMFOJS \"OLBSBhOO ZMOEBLJ OÐGVTV PMVQ PSUBMBNBOÐGVTBSUõI[ZÐ[EF PMBSBLIFTBQ- MBONõUS #V ÀFLJSHF LF[ TÀSBELUBO TPOSB CBöMBEô #VOB HÌSF \"OLBSBhOO ZMOEB CFLMFOFO OPLUBEBOFOGB[MBLBÀNFUSFV[BLMBöNöUS OÑGVTVZBLMBöLPMBSBLLBÀUS (e0,36 = 1,4) \" # $ \" # $ % & % & 1. E 2. D 48 3. D 4. A
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