Roboguru

Jika b−cloga​=c−alogb​=a−blogc​, tentukan nilai aabbcc!

Pertanyaan

Jika bcloga=calogb=ablogc, tentukan nilai aabbcc

Pembahasan Soal:

Ingat 

  • alogbc=alogb+alogc  
  • alogbm=malogb   
  • alogf(x)=bf(x)=ab 

Perhatikan perhitungan berikut 

Misalnya 

bcloga=calogb=ablogc=k  

Maka 

bclogalogaalogalogaacalogblogbblogblogbbablogclogcclogclogcc============kk(bc)ak(bc)ak(bc)kk(ca)bk(ca)=bk(ca)kk(ab)ck(ab)ck(ab) 

Selanjutnya 

logaa+logbb+logccak(bc)+bk(ca)+ck(ab)abkack+bckabk+ackbck0aabbccaabbcc======log(aabbcc)log(aabbcc)log(aabbcc)log(aabbcc)1001 

Dengan demikian, aabbcc=1

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Lestari

Mahasiswa/Alumni Universitas Sebelas Maret

Terakhir diupdate 13 September 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

Membantu

Kurang Membantu

Apakah pembahasan ini membantu?

Belum menemukan yang kamu cari?

Post pertanyaanmu ke Tanya Jawab, yuk

Mau Bertanya

Pertanyaan yang serupa

Nilai yang memenuhi persamaan  adalah ...

Pembahasan Soal:

Ingat bentuk umum atau definisi dari logaritma yaitu scriptbase log invisible function application b end scriptbase presuperscript a equals c rightwards arrow a to the power of c equals b dan sifat-sifat pada logaritma yaitu

  • scriptbase log invisible function application a end scriptbase presuperscript a equals 1
  • scriptbase log invisible function application b end scriptbase presuperscript a minus scriptbase log invisible function application c end scriptbase presuperscript a equals scriptbase log invisible function application b over c end scriptbase presuperscript a
  • scriptbase log invisible function application b end scriptbase presuperscript a plus scriptbase log invisible function application c end scriptbase presuperscript a equals scriptbase log invisible function application b c end scriptbase presuperscript a
  • scriptbase log invisible function application b to the power of m end scriptbase presuperscript a equals m times scriptbase log invisible function application b end scriptbase presuperscript a

serta syarat basis pada logaritma scriptbase log invisible function application b end scriptbase presuperscript a yaitu a greater than 0 dan a not equal to 1. Sehingga 

table attributes columnalign right center left columnspacing 0px end attributes row cell scriptbase log invisible function application open parentheses 2 x plus 5 close parentheses end scriptbase presuperscript x plus 1 end presuperscript end cell equals cell fraction numerator scriptbase log squared invisible function application 24 end scriptbase presuperscript 2 minus scriptbase log squared invisible function application 6 end scriptbase presuperscript 2 over denominator 2 times scriptbase log invisible function application 12 end scriptbase presuperscript 2 end fraction end cell row blank equals cell fraction numerator left parenthesis scriptbase log invisible function application 24 end scriptbase presuperscript 2 plus scriptbase log invisible function application 6 end scriptbase presuperscript 2 right parenthesis left parenthesis scriptbase log invisible function application 24 end scriptbase presuperscript 2 minus scriptbase log invisible function application 6 end scriptbase presuperscript 2 right parenthesis over denominator scriptbase log invisible function application 12 squared end scriptbase presuperscript 2 end fraction end cell row blank equals cell fraction numerator open parentheses scriptbase log invisible function application left parenthesis 24 times 6 right parenthesis end scriptbase presuperscript 2 close parentheses open parentheses scriptbase log invisible function application 24 over 6 end scriptbase presuperscript 2 close parentheses over denominator scriptbase log invisible function application 144 end scriptbase presuperscript 2 end fraction end cell row blank equals cell fraction numerator open parentheses scriptbase log invisible function application 144 end scriptbase presuperscript 2 close parentheses open parentheses scriptbase log invisible function application 4 end scriptbase presuperscript 2 close parentheses over denominator scriptbase log invisible function application 144 end scriptbase presuperscript 2 end fraction end cell row blank equals cell scriptbase log invisible function application 4 end scriptbase presuperscript 2 end cell row blank equals cell scriptbase log invisible function application 2 squared end scriptbase presuperscript 2 end cell row blank equals cell 2 times scriptbase log invisible function application 2 end scriptbase presuperscript 2 end cell row blank equals 2 end table

Dari hasil di atas didapatkan scriptbase log invisible function application open parentheses 2 x plus 5 close parentheses end scriptbase presuperscript x plus 1 end presuperscript equals 2, sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell scriptbase log invisible function application open parentheses 2 x plus 5 close parentheses end scriptbase presuperscript x plus 1 end presuperscript end cell equals 2 row cell left parenthesis x plus 1 right parenthesis squared end cell equals cell 2 x plus 5 end cell row cell x squared plus 2 x plus 1 end cell equals cell 2 x plus 5 end cell row cell x squared plus 2 x minus 2 x plus 1 minus 5 end cell equals 0 row cell x squared minus 4 end cell equals 0 row cell x squared end cell equals 4 row x equals cell square root of 4 end cell row x equals cell plus-or-minus 2 end cell end table

Didapatkan hasil  nilai x yaitu negative 2 atau 2. Karena syarat basis dari logaritma scriptbase log invisible function application b end scriptbase presuperscript a yaitu a greater than 0 dan a not equal to 1 maka nilai x yang memenuhi adalah x equals 2.

Jadi, dapat disimpulkan bahwa nilai x yang memenuhi persamaan scriptbase log invisible function application left parenthesis 2 x plus 5 right parenthesis end scriptbase presuperscript x plus 1 end presuperscript equals fraction numerator scriptbase log squared invisible function application 24 end scriptbase presuperscript 2 minus scriptbase log squared invisible function application 6 end scriptbase presuperscript 2 over denominator 2 times scriptbase log invisible function application 12 end scriptbase presuperscript 2 end fraction adalah x equals 2.

Oleh karena itu, jawaban yang benar adalah B.

0

Roboguru

Jika , tentukanlah .

Pembahasan Soal:

Untuk meyelesaikan soal perlu dilakukan permisalan sebagai berikut.

Misal fraction numerator log invisible function application a over denominator b minus c end fraction equals fraction numerator log invisible function application b over denominator c minus a end fraction equals fraction numerator log invisible function application c over denominator a minus b end fraction equals k sehingga akan didapatkan

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator log invisible function application a over denominator b minus c end fraction end cell equals k row cell log invisible function application a end cell equals cell k left parenthesis b minus c right parenthesis end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator log invisible function application b over denominator c minus a end fraction end cell equals k row cell log invisible function application b end cell equals cell k left parenthesis c minus a right parenthesis end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator log invisible function application c over denominator a minus b end fraction end cell equals k row cell log invisible function application c end cell equals cell k left parenthesis a minus b right parenthesis end cell end table

Ingat sifat pada bentuk logaritma yaitu scriptbase log invisible function application b end scriptbase presuperscript a plus scriptbase log invisible function application c end scriptbase presuperscript a equals scriptbase log invisible function application c end scriptbase presuperscript a dan scriptbase log invisible function application b to the power of m end scriptbase presuperscript a equals m times scriptbase log invisible function application b end scriptbase presuperscript a maka

table attributes columnalign right center left columnspacing 0px end attributes row cell log invisible function application left parenthesis a to the power of a times b to the power of b times c to the power of c right parenthesis end cell equals cell log invisible function application a to the power of a plus log invisible function application b to the power of b plus log invisible function application c to the power of c end cell row blank equals cell a times log invisible function application a plus b times log invisible function application b plus c times log invisible function application c end cell row blank equals cell a times k left parenthesis b minus c right parenthesis plus b times k left parenthesis c minus a right parenthesis plus c times k left parenthesis a minus b right parenthesis end cell row blank equals cell a times left parenthesis k b minus k c right parenthesis plus b times left parenthesis k c minus k a right parenthesis plus c times left parenthesis k a minus k b right parenthesis end cell row blank equals cell left parenthesis k a b minus k a c right parenthesis plus left parenthesis k b c minus k a b right parenthesis plus left parenthesis k a c minus k b c right parenthesis end cell row blank equals cell k a b minus k a c plus k b c minus k a b plus k a c minus k b c end cell row blank equals 0 end table

Didapatkan log invisible function application left parenthesis a to the power of a times b to the power of b times c to the power of c right parenthesis equals 0, ingat sifat pada logaritma yaitu scriptbase log invisible function application b end scriptbase presuperscript a equals scriptbase log invisible function application c end scriptbase presuperscript a maka b equals c sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell log invisible function application open parentheses a to the power of a times b to the power of b times c to the power of c close parentheses end cell equals 0 row cell log invisible function application left parenthesis a to the power of a times b to the power of b times c to the power of c right parenthesis end cell equals cell log invisible function application 1 end cell row cell a to the power of a times b to the power of b times c to the power of c end cell equals 1 end table

Jadi, dapat disimpulkan bahwa hasil dari a to the power of a times b to the power of b times c to the power of c adalah 1.

0

Roboguru

Tentukan penyelesaian dari sistem persamaan berikut: a. {5logx+5logy=55logx4−5logy3=−1​

Pembahasan Soal:

Ingat kembali:

alogb+alogc=alogbcalogbalogc=alogcbalogbm=m×alogbaloga=1anam=amnaman=am+n 

Diberikan sistem persamaan:

{5logx+5logy=55logx45logy3=1

Maka:

  • Buat persamaan (1)

5logx+5logy5logx+5logy5logx+5logy5logxyxyx======55155log55log5555y55 

  • Buat persamaan (2)

5logx45logy35logx45logy35logx45logy35logx45logy35logy3x4y3x4======11115log55log515log5151 

  • Substitusi persamaan (1) ke (2)

y3x4y31y3y3y3y4y7yyy=========51x45(y55)45y452055205521572153125 

  • Subtitusi nilai y ke (1)

x====y5553555225 

Jadi, penyelesaian dari sistem persamaan {5logx+5logy=55logx45logy3=1 adalah x=25 dan y=125.

0

Roboguru

Jumlah akar-akar persamaan:  sama dengan ...

Pembahasan Soal:

Ingat bentuk umum atau definisi dari logaritma yaitu scriptbase log invisible function application b end scriptbase presuperscript a equals c rightwards arrow a to the power of c equals b dan Ingat sifat-sifat pada bentuk logaritma yaitu 

  • scriptbase log invisible function application a end scriptbase presuperscript a equals 1
  • scriptbase log invisible function application b to the power of m end scriptbase presuperscript a equals m times scriptbase log invisible function application b end scriptbase presuperscript a
  • scriptbase log invisible function application b end scriptbase presuperscript a plus scriptbase log invisible function application c end scriptbase presuperscript a equals scriptbase log invisible function application b c end scriptbase presuperscript a

Untuk menyelesaikan persamaan tersebut maka dilakkukan permisalan terlebih dahulu.

Misalkan 3 to the power of x equals y sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 times 3 to the power of x plus 18 over 3 to the power of x minus 14 end cell equals 0 row cell 2 y plus 18 over y minus 14 end cell equals 0 row cell 2 y squared plus 18 minus 14 y end cell equals 0 row cell 2 y squared minus 14 y plus 18 end cell equals 0 row cell y squared minus 7 y plus 9 end cell equals 0 end table

Untuk menentukan akar-akar persamaan di atas maka dapat menggunakan rumus y equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction.

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction end cell row blank equals cell fraction numerator negative left parenthesis negative 7 right parenthesis plus-or-minus square root of left parenthesis negative 7 right parenthesis squared minus 4 times 1 times 9 end root over denominator 2 times 1 end fraction end cell row blank equals cell fraction numerator 7 plus-or-minus square root of 49 minus 36 end root over denominator 2 end fraction end cell row blank equals cell fraction numerator 7 plus-or-minus square root of 13 over denominator 2 end fraction end cell end table

Didapatkan y subscript 1 equals fraction numerator 7 plus square root of 13 over denominator 2 end fraction dan y subscript 2 equals fraction numerator 7 minus square root of 13 over denominator 2 end fraction. Sehingga 

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 to the power of x end cell equals y row cell 3 to the power of x end cell equals cell fraction numerator 7 plus square root of 13 over denominator 2 end fraction end cell row x equals cell scriptbase log invisible function application fraction numerator 7 plus square root of 13 over denominator 2 end fraction end scriptbase presuperscript 3 end cell row cell x subscript 1 end cell equals cell scriptbase log invisible function application fraction numerator 7 plus square root of 13 over denominator 2 end fraction end scriptbase presuperscript 3 end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 to the power of x end cell equals y row cell 3 to the power of x end cell equals cell fraction numerator 7 minus square root of 13 over denominator 2 end fraction end cell row x equals cell scriptbase log invisible function application fraction numerator 7 minus square root of 13 over denominator 2 end fraction end scriptbase presuperscript 3 end cell row cell x subscript 1 end cell equals cell scriptbase log invisible function application fraction numerator 7 minus square root of 13 over denominator 2 end fraction end scriptbase presuperscript 3 end cell end table

Menentukan x subscript 1 plus x subscript 2

table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 plus x subscript 2 end cell equals cell scriptbase log invisible function application fraction numerator 7 plus square root of 13 over denominator 2 end fraction end scriptbase presuperscript 3 plus scriptbase log invisible function application fraction numerator 7 minus square root of 13 over denominator 2 end fraction end scriptbase presuperscript 3 end cell row blank equals cell scriptbase log invisible function application open parentheses open parentheses fraction numerator 7 plus square root of 13 over denominator 2 end fraction close parentheses open parentheses fraction numerator 7 minus square root of 13 over denominator 2 end fraction close parentheses close parentheses end scriptbase presuperscript 3 end cell row blank equals cell scriptbase log invisible function application open parentheses fraction numerator open parentheses 7 plus square root of 13 close parentheses open parentheses 7 minus square root of 13 close parentheses over denominator 2 times 2 end fraction close parentheses end scriptbase presuperscript 3 end cell row blank equals cell scriptbase log invisible function application open parentheses fraction numerator 49 plus 7 square root of 13 minus 7 square root of 13 minus 13 over denominator 4 end fraction close parentheses end scriptbase presuperscript 3 end cell row blank equals cell scriptbase log invisible function application open parentheses fraction numerator 49 minus 13 over denominator 4 end fraction close parentheses end scriptbase presuperscript 3 end cell row blank equals cell scriptbase log invisible function application open parentheses 36 over 4 close parentheses end scriptbase presuperscript 3 end cell row blank equals cell scriptbase log invisible function application 9 end scriptbase presuperscript 3 end cell row blank equals cell scriptbase log invisible function application 3 squared end scriptbase presuperscript 3 end cell row blank equals cell 2 times scriptbase log invisible function application 3 end scriptbase presuperscript 3 end cell row blank equals cell 2 times 1 end cell row blank equals 2 end table

Jadi, dapat disimpulkan bahwa jumlah akar-akar persamaan: 2 times 3 to the power of x plus 18 over 3 to the power of x minus 14 equals 0 sama dengan 2.

Oleh karena itu, jawaban yang benar adalah D.

0

Roboguru

Jumlah semua bilangan bulat x sehingga 2log(x2−4x−1) merupakan bilangan bulat adalah ...

Pembahasan Soal:

Ingat 

alogf(x)=bf(x)=ab 

Perhatikan perhitungan berikut 

Misalnya 

2log(x24x1)=kk bilangan bulat

Maka 

 table attributes columnalign right center left columnspacing 0px end attributes row cell blank squared log space open parentheses x squared minus 4 x minus 1 close parentheses end cell equals k row cell open parentheses x squared minus 4 x minus 1 close parentheses end cell equals cell 2 to the power of k end cell row cell x squared minus 4 x end cell equals cell 2 to the power of k plus 1 end cell row cell x squared minus 4 x plus 4 end cell equals cell 2 to the power of k plus 1 plus 4 space end cell row cell open parentheses x minus 2 close parentheses squared end cell equals cell 2 to the power of k plus 5 end cell end table 

Untuk k<0, maka 2k+5 tidak bulat sehingga tidak ada x tidak bilangan bulat yang memenuhi. 

Untuk k=0 

(x2)2(x2)2(x2)2===20+51+56 

Tidak ada bilangan bulat x yang memenuhi persamaan tersebut. 


Untuk k=1  

(x2)2(x2)2(x2)2===21+52+57 

Tidak ada bilangan bulat x yang memenuhi persamaan tersebut. 


Untuk k=2   

(x2)2(x2)2(x2)2x2x1=====22+54+593ataux2=35x2=1 

Karena tidak ada 2k+5 lainnya yang merupakan kuadrat sempurna, maka tidak ada nilai x lainnya yang memenuhi persamaan tersebut. 

Jadi, nilai x yang memenuhi persamaan tersebut adalah x1=5danx2=1 

x1+x2==5+(1)4 

Dengan demikian, jumlah semua nilai  x yang memenuhi persamaan tersebut adalah 4

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

Membantu

Kurang Membantu

Apakah pembahasan ini membantu?

Belum menemukan yang kamu cari?

Post pertanyaanmu ke Tanya Jawab, yuk

Mau Bertanya

RUANGGURU HQ

Jl. Dr. Saharjo No.161, Manggarai Selatan, Tebet, Kota Jakarta Selatan, Daerah Khusus Ibukota Jakarta 12860

Coba GRATIS Aplikasi Ruangguru

Produk Ruangguru

Produk Lainnya

Hubungi Kami

Ikuti Kami

©2021 Ruangguru. All Rights Reserved