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Jika  dan , tentukan nilai logaritma berikut. c.

Pertanyaan

Jika begin mathsize 14px style log space 2 equals m end style dan begin mathsize 14px style log space 3 equals n end style, tentukan nilai logaritma berikut.

c. begin mathsize 14px style log space 8 over 3 end style 

Pembahasan Soal:

Ingat sifat logaritma berikut :

alogcbalogbm==alogbalogcmalogb

Diketahui  begin mathsize 14px style log space 2 equals m end style dan begin mathsize 14px style log space 3 equals n end style, sehingga bentuk logaritma di atas dapat dihitung seperti berikut :

begin mathsize 14px style log 8 over 3 equals log space 8 minus log space 3 space space space space space space space space space equals log space 2 cubed minus log space 3 space space space space space space space space space equals 3 space log space 2 minus log space 3 space space space space space space space space space equals 3 m minus n end style

Dengan demikian, nilai dari begin mathsize 14px style log space 8 over 3 end style adalah 3mn.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

I. Kumaralalita

Mahasiswa/Alumni Universitas Gadjah Mada

Terakhir diupdate 30 Agustus 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.

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Roboguru

Hasil dari  adalah ...

Pembahasan Soal:

Ingat!

table attributes columnalign right center left columnspacing 0px end attributes row cell log subscript a space b times log subscript b space c end cell equals cell log subscript a space c end cell row cell log subscript a space b to the power of n end cell equals cell n log subscript a space b end cell row cell log subscript a space b over c end cell equals cell log subscript a space b minus log subscript a space c end cell end table 

Maka, hasil dari fraction numerator log subscript 5 space square root of 32 times log subscript 2 space 9 times log subscript 3 space begin display style 1 over 25 end style over denominator log subscript square root of 2 end subscript space 28 minus log subscript square root of 2 end subscript space 7 end fraction adalah:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator log subscript 5 space square root of 32 times log subscript 2 space 9 times log subscript 3 space begin display style 1 over 25 end style over denominator log subscript square root of 2 end subscript space 28 minus log subscript square root of 2 end subscript space 7 end fraction end cell row blank equals cell fraction numerator log subscript 5 space square root of 2 to the power of 5 end root times log subscript 2 space 3 squared times log subscript 3 space begin display style 1 over 5 squared end style over denominator log subscript square root of 2 end subscript begin display style fraction numerator space 28 over denominator 7 end fraction end style end fraction end cell row blank equals cell fraction numerator log subscript 5 space 2 to the power of begin display style 5 over 2 end style end exponent times 2 times log subscript 2 space 3 times log subscript 3 space begin display style 5 to the power of negative 2 end exponent end style over denominator 4 end fraction end cell row blank equals cell fraction numerator begin display style 5 over 2 end style times 2 times open parentheses negative 2 close parentheses log subscript 5 space 2 times log subscript 2 space 3 times log subscript 3 space begin display style 5 end style over denominator 4 end fraction end cell row blank equals cell fraction numerator begin display style negative 10 times log subscript 5 space 5 end style over denominator 4 end fraction end cell row blank equals cell fraction numerator begin display style negative 10 times 1 end style over denominator 4 end fraction end cell row blank equals cell negative 5 over 2 end cell end table 

Sehingga, table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator log subscript 5 space square root of 32 times log subscript 2 space 9 times log subscript 3 space begin display style 1 over 25 end style over denominator log subscript square root of 2 end subscript space 28 minus log subscript square root of 2 end subscript space 7 end fraction end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 5 over 2 end cell end table.

Jadi, jawaban yang tepat adalah C.

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Roboguru

Hasil dari  adalah ...

Pembahasan Soal:

Gunakan sifat-sifat logaritma untuk mengerjakan soal tersebut.


table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator log presuperscript 3 space 25 times log presuperscript 5 space 81 minus log presuperscript 4 space 2 over denominator log presuperscript 3 space 36 minus log presuperscript 3 space 4 end fraction end cell row blank equals cell fraction numerator log presuperscript 3 space 5 squared times log presuperscript 5 space 3 to the power of 4 minus log presuperscript 2 squared end presuperscript space 2 over denominator log presuperscript 3 space open parentheses begin display style 36 over 4 end style close parentheses end fraction end cell row blank equals cell fraction numerator 2 times 4 times log presuperscript 3 space 5 times log presuperscript 5 space 3 minus begin display style 1 half end style times log presuperscript 2 space 2 over denominator log presuperscript 3 space 9 end fraction end cell row blank equals cell fraction numerator 8 times log presuperscript 3 space 3 minus begin display style 1 half end style over denominator log presuperscript 3 space 3 squared end fraction end cell row blank equals cell fraction numerator 8 minus begin display style 1 half end style over denominator 2 times log presuperscript 3 space 3 end fraction end cell row blank equals cell fraction numerator begin display style 15 over 2 end style over denominator 2 end fraction end cell row blank equals cell 15 over 4 end cell end table


Oleh karena itu, jawaban yang benar adalah B.

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Roboguru

...

Pembahasan Soal:

Ingatlah sifat-sifat logaritma berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell log space straight a to the power of straight n end cell equals cell straight n times log space straight a end cell row cell log space straight a over straight b end cell equals cell log space straight a minus log space straight b end cell row cell log space ab end cell equals cell log space straight a plus log space straight b end cell row cell scriptbase log space 1 end scriptbase presuperscript straight a end cell equals 0 end table

Diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 open parentheses log presuperscript 2 space y close parentheses minus log presuperscript 2 space y squared plus log presuperscript 2 space 1 over y end cell equals cell scriptbase log space y cubed end scriptbase presuperscript 2 minus log presuperscript 2 space y squared plus log presuperscript 2 space 1 over y end cell row blank equals cell log presuperscript 2 space y cubed over y squared plus log presuperscript 2 space 1 over y end cell row blank equals cell log presuperscript 2 space y plus log presuperscript 2 space 1 over y end cell row blank equals cell log presuperscript 2 space open parentheses y cross times 1 over y close parentheses end cell row blank equals cell log presuperscript 2 space 1 end cell row blank equals 0 end table

Maka, nilai dari 3 open parentheses log presuperscript 2 space y close parentheses minus log presuperscript 2 space y squared plus log presuperscript 2 space 1 over y equals 0.

0

Roboguru

Dengan menggunakan tabel atau kalkulator, hitunglah nilai  berikut. c.

Pembahasan Soal:

begin mathsize 14px style x equals 10 to the power of 0 comma 699 minus 1 end exponent space space space equals 10 to the power of negative 0 comma 301 end exponent space space space almost equal to 0 comma 500035 end style

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