Roboguru

Sederhanakan: b. 2log8​+16log41​+4log64

Pertanyaan

Sederhanakan:

b. 2log8+16log41+4log64  

Pembahasan Soal:

Ingat kembali sifat-sifat logaritma:

alogbmanlogbmaloga===malogbnmalogb1 

dan ingat kembali sifat eksponen:

am1=am 

Sehingga

2log8+16log41+4log64========2log23+42log41+4log432log223+42log41+4log43232log2+(21)4log4+34log4231+(21)1+312321+32321+26284   

Dengan demikian hasil dari 2log8+16log41+4log64 adalah 4.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Amamah

Mahasiswa/Alumni Universitas Negeri Malang

Terakhir diupdate 13 September 2021

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Bentuk sederhana dari  = ...

Pembahasan Soal:

Sifat-sifat logaritma:

  • begin mathsize 12px style a to the power of n log b equals 1 over n times log presuperscript a b end style
  • begin mathsize 12px style log presuperscript a b equals fraction numerator log b over denominator log a end fraction end style
  • begin mathsize 12px style log presuperscript a b minus log presuperscript a c equals log presuperscript a open parentheses b over c close parentheses end style
  • begin mathsize 12px style log presuperscript a b to the power of m equals m times log presuperscript a b end style
  • begin mathsize 12px style log presuperscript a a equals 1 end style

begin mathsize 12px style fraction numerator log presuperscript square root of 3 end presuperscript 100 minus log 9 minus log presuperscript 5 625 over denominator log presuperscript 2 12 minus log presuperscript 2 3 end fraction  equals fraction numerator log presuperscript open parentheses 3 close parentheses to the power of 1 half end exponent end presuperscript 100 times log 9 minus log presuperscript 5 5 to the power of 4 over denominator log presuperscript 2 open parentheses begin display style 12 over 3 end style close parentheses end fraction  equals fraction numerator 2 times log presuperscript 3 100 times log 9 minus 4 times log presuperscript 5 5 over denominator log presuperscript 2 4 end fraction  equals fraction numerator log presuperscript 3 100 squared times log 9 minus 4 over denominator log presuperscript 2 2 squared end fraction  equals fraction numerator begin display style fraction numerator log 10 to the power of 4 times log 9 over denominator log 3 end fraction end style minus 4 over denominator 2 times log presuperscript 2 2 end fraction  equals fraction numerator log 10 to the power of 4 times log presuperscript 3 9 minus 4 over denominator 2 end fraction  equals fraction numerator log 10 to the power of 4 times log presuperscript 3 3 squared minus 4 over denominator 2 end fraction  equals fraction numerator 4 times 2 times log presuperscript 3 3 minus 4 over denominator 2 end fraction  equals fraction numerator 8 minus 4 over denominator 2 end fraction  equals 4 over 2  equals 2 end style

Jadi, begin mathsize 12px style fraction numerator log presuperscript square root of 3 end presuperscript 100 minus log 9 minus log presuperscript 5 625 over denominator log presuperscript 2 12 minus log presuperscript 2 3 end fraction end style = 2.

0

Roboguru

Tentukan nilai dari: a. 25log43 b. 9log25 c.

Pembahasan Soal:

Sifat-sifat bentuk logaritma:

amlogbn=mnalogb ; alogbm=malogbaloga=1

Penyelesaian:

a. 25log43

25log43=====532log4532log22532(2log2)56156

Jadi, 25log43=56.


b. 9log25

9log25====32log5222(3log5)1(3log5)3log5

Jadi, 9log25=3log5.


c. space presuperscript square root of 3 end presuperscript log space 81

table attributes columnalign right center left columnspacing 0px end attributes row cell space presuperscript square root of 3 end presuperscript log space 81 end cell equals cell log presuperscript 3 to the power of 1 half end exponent end presuperscript space 3 to the power of 4 end cell row blank equals cell fraction numerator 4 over denominator begin display style 1 half end style end fraction space open parentheses log presuperscript 3 space 3 close parentheses end cell row blank equals cell 4 times 2 open parentheses 1 close parentheses end cell row blank equals 8 end table

Jadi, space presuperscript square root of 3 end presuperscript log space 81 equals 8.

0

Roboguru

Nilai dari  adalah ....

Pembahasan Soal:

begin mathsize 14px style open square brackets fraction numerator log presuperscript 3 space 2. log presuperscript 4 space 27 plus log presuperscript 3 space 81 over denominator log presuperscript 2 space 8 minus log presuperscript 2 space 4 end fraction close square brackets squared end style

begin mathsize 14px style equals open parentheses fraction numerator log presubscript space presuperscript 3 invisible function application space 2. space log presubscript space presuperscript 2 squared end presuperscript invisible function application space 3 cubed plus log presubscript space presuperscript 3 space invisible function application 3 to the power of 4 over denominator log presubscript space presuperscript 2 space invisible function application open parentheses 8 over 4 close parentheses end fraction close parentheses squared  equals open parentheses fraction numerator 3 over 2. log presubscript space presuperscript 3 space invisible function application 2. space log presubscript space presuperscript 2 space invisible function application 3 plus 4. log presubscript space presuperscript 3 space invisible function application 3 over denominator log presubscript space presuperscript 2 space invisible function application 2 end fraction close parentheses squared  equals open parentheses fraction numerator 3 over 2 space. log presubscript space presuperscript 3 space invisible function application 3 plus 4 over denominator 1 end fraction close parentheses squared  equals open parentheses 3 over 2 plus 4 close parentheses squared  equals open parentheses 11 over 2 close parentheses squared  equals 121 over 4 end style

0

Roboguru

Tentukan nilai logaritma berikut.  c.

Pembahasan Soal:

Ingat sifat bentuk logaritma:

table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript a to the power of m end presuperscript space b to the power of n end cell equals cell n over m times log presuperscript a space b end cell row cell log presuperscript a space b to the power of n end cell equals cell n times log presuperscript a space b end cell row cell log presuperscript a space a end cell equals 1 row cell log presuperscript a space b times log presuperscript b space c end cell equals cell log presuperscript a space c comma space b not equal to 1 end cell end table  

sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript 2 space 9 cross times log presuperscript 3 space 16 plus log presuperscript 4 space 32 end cell equals cell log presuperscript 2 space 3 squared cross times log presuperscript 3 space 2 to the power of 4 plus log presuperscript 2 squared end presuperscript space 2 to the power of 5 end cell row blank equals cell 2 times log presuperscript 2 space 3 cross times log presuperscript 3 space 2 to the power of 4 plus 5 over 2 times log presuperscript 2 space 2 end cell row blank equals cell 2 times log presuperscript 2 space 2 to the power of 4 plus 5 over 2 times 1 end cell row blank equals cell 2 times 4 times log presuperscript 2 space 2 plus 5 over 2 end cell row blank equals cell 8 times 1 plus 5 over 2 end cell row blank equals cell fraction numerator 16 plus 5 over denominator 2 end fraction end cell row blank equals cell 21 over 2 end cell end table  

Dengan demikian nilai dari log presuperscript 2 space 9 cross times log presuperscript 3 space 16 plus log presuperscript 4 space 32 adalah 21 over 2

0

Roboguru

Hasil dari  adalah...

Pembahasan Soal:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator blank cubed log space 36 times to the power of 6 log space 81 plus to the power of 4 log space 32 over denominator blank to the power of begin display style 1 over 9 end style end exponent log space 27 end fraction end cell equals cell fraction numerator blank cubed log space 6 squared times to the power of 6 log space 3 to the power of 4 plus to the power of 2 squared end exponent log space 2 to the power of 5 over denominator blank to the power of begin display style 3 to the power of negative 2 end exponent end style end exponent log space 3 cubed end fraction end cell row blank equals cell fraction numerator 2 space cubed log space 6 times to the power of 6 log space 3 to the power of 4 plus begin display style 5 over 2 end style space squared log space 2 over denominator negative begin display style 3 over 2 end style space cubed log space 3 end fraction end cell row blank equals cell fraction numerator 2 space cubed log space 3 to the power of 4 plus begin display style 5 over 2 times 1 end style over denominator negative begin display style 3 over 2 end style times 1 end fraction end cell row blank equals cell fraction numerator 2 times 4 space cubed log space 3 plus begin display style 5 over 2 end style over denominator negative begin display style 3 over 2 end style end fraction end cell row blank equals cell fraction numerator 2 times 4 space plus begin display style 5 over 2 end style over denominator negative begin display style 3 over 2 end style end fraction end cell row blank equals cell fraction numerator 8 plus begin display style 5 over 2 end style over denominator negative begin display style 3 over 2 end style end fraction end cell row blank equals cell fraction numerator begin display style 21 over 2 end style over denominator negative begin display style 3 over 2 end style end fraction end cell row blank equals cell 21 over 2 cross times left parenthesis negative 2 over 3 right parenthesis end cell row blank equals cell negative 7 end cell end table end style

jadi, hasil dari persamaan tersebut adalah negative 7 

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