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Pertanyaan

Hasil kali akar-akar persamaan: scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 2 plus fraction numerator 6 over denominator scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 2 end fraction equals scriptbase log invisible function application 32 end scriptbase presuperscript 2 sama dengan ...

  1. 32

  2. 28

  3. 21

  4. 12

  5. 6

Pembahasan Soal:

Ingat materi persamaan logaritma dengan basisnya adalah fungsi yang sama dan numerusnya adalah fungsi yang sama dan 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. Ingat juga sifat-sifat pada bentuk logaritma yaitu scriptbase log invisible function application a end scriptbase presuperscript a equals 1 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.

Untuk menyelesaikan persamaan tersebut maka dilakkukan permisalan terlebih dahulu.

Misalkan scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 2 equals y sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell scriptbase log invisible function application open parentheses x plus 1 close parentheses end scriptbase presuperscript 2 plus fraction numerator 6 over denominator scriptbase log invisible function application open parentheses x plus 1 close parentheses end scriptbase presuperscript 2 end fraction end cell equals cell scriptbase log invisible function application 32 end scriptbase presuperscript 2 end cell row cell y plus 6 over y end cell equals cell scriptbase log invisible function application 2 to the power of 5 end scriptbase presuperscript 2 end cell row cell y plus 6 over y end cell equals cell 5 times scriptbase log invisible function application 2 end scriptbase presuperscript 2 end cell row cell y plus 6 over y end cell equals 5 row cell y squared plus 6 end cell equals cell 5 y end cell row cell y squared minus 5 y plus 6 end cell equals 0 row cell left parenthesis y minus 2 right parenthesis left parenthesis y minus 3 right parenthesis end cell equals 0 row cell y minus 2 end cell equals cell 0 blank atau blank y minus 3 equals 0 end cell row y equals cell 2 blank atau blank y equals 3 end cell end table

Didapatkan hasil y subscript 1 equals 2 dan y subscript 2 equals 3, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell y subscript 2 end cell equals 3 row cell scriptbase log space invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 2 end cell equals 2 row cell 2 squared end cell equals cell x plus 1 end cell row 4 equals cell x plus 1 end cell row cell 4 minus 1 end cell equals x row 3 equals x end table

table attributes columnalign right center left columnspacing 0px end attributes row cell y subscript 2 end cell equals 3 row cell scriptbase log space invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 2 end cell equals 3 row cell 2 cubed end cell equals cell x plus 1 end cell row 8 equals cell x plus 1 end cell row cell 8 minus 1 end cell equals x row 7 equals x end table

Didapatkan hasil x subscript 1 equals 3 dan x subscript 2 equals 7, sehingga hasil akar-akarnya adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 times x subscript 2 end cell equals cell 3 times 7 end cell row blank equals 21 end table

Jadi, dapat disimpulkan bahwa hasil kali akar-akar persamaan: scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 2 plus fraction numerator 6 over denominator scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 2 end fraction equals scriptbase log invisible function application 32 end scriptbase presuperscript 2 sama dengan 21.

Oleh karena itu, jawaban yang benar adalah C.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

M. Iqbal

Mahasiswa/Alumni Universitas Negeri Semarang

Terakhir diupdate 11 Juli 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Jumlah akar-akar persamaan:  sama dengan ...

Pembahasan Soal:

Ingat materi persamaan logaritma dengan basisnya adalah fungsi yang sama dan numerusnya adalah fungsi yang sama dan 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. Ingat juga sifat-sifat pada bentuk logaritma yaitu scriptbase log invisible function application a end scriptbase presuperscript a equals 1 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.

Untuk menyelesaikan persamaan tersebut maka dilakkukan permisalan terlebih dahulu.

Misalkan scriptbase log invisible function application x end scriptbase presuperscript 2 equals y, sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell scriptbase log invisible function application x end scriptbase presuperscript 2 open parentheses scriptbase log invisible function application x end scriptbase presuperscript 2 minus 3 close parentheses end cell equals cell scriptbase log invisible function application 16 end scriptbase presuperscript 2 end cell row cell y open parentheses y minus 3 close parentheses end cell equals cell scriptbase log invisible function application 2 to the power of 4 end scriptbase presuperscript 2 end cell row cell y squared minus 3 y end cell equals cell 4 times scriptbase log invisible function application 2 end scriptbase presuperscript 2 end cell row cell y squared minus 3 y end cell equals 4 row cell y squared minus 3 y minus 4 end cell equals 0 row cell left parenthesis y minus 4 right parenthesis left parenthesis y plus 1 right parenthesis end cell equals 0 row cell y minus 4 end cell equals cell 0 blank atau blank y plus 1 equals 0 end cell row y equals cell 4 blank atau blank y equals negative 1 end cell end table

 Didapatkan hasil y subscript 1 equals 4 dan y subscript 2 equals negative 1, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell y subscript 1 end cell equals 4 row cell scriptbase log invisible function application x end scriptbase presuperscript 2 end cell equals 4 row cell 2 to the power of 4 end cell equals x row 16 equals x end table

table attributes columnalign right center left columnspacing 0px end attributes row cell y subscript 2 end cell equals cell negative 1 end cell row cell scriptbase log invisible function application x end scriptbase presuperscript 2 end cell equals cell negative 1 end cell row cell 2 to the power of negative 1 end exponent end cell equals x row cell 1 half end cell equals x end table

Didapatkan hasil x subscript 1 equals 16 dan x subscript 2 equals 1 half, sehingga jumlahnya adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 plus x subscript 2 end cell equals cell 16 plus 1 half end cell row blank equals cell 16 plus 0 , 5 end cell row blank equals cell 16 , 5 end cell end table

Jadi, dapat disimpulkan bahwa jumlah akar-akar persamaan: scriptbase log invisible function application x end scriptbase presuperscript 2 open parentheses scriptbase log invisible function application x end scriptbase presuperscript 2 minus 3 close parentheses equals scriptbase log invisible function application 16 end scriptbase presuperscript 2 sama dengan 16 , 5.

Oleh karena itu, jawaban yang benar adalah A.

0

Roboguru

Misal persamaan logaritma:  mempunyai akar-akar  dan  maka

Pembahasan Soal:

Ingat materi persamaan logaritma dengan basisnya adalah fungsi yang berbeda dan numerusnya adalah fungsi yang sama dan 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.

Ingat juga 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 to the power of m end scriptbase presuperscript a to the power of n end presuperscript equals m over n times scriptbase log invisible function application b end scriptbase presuperscript a

Untuk menyelesaikan persamaan tersebut maka dilakkukan permisalan terlebih dahulu.

Misalkan scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 3 equals y sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell scriptbase log invisible function application open parentheses x plus 1 close parentheses end scriptbase presuperscript 3 plus fraction numerator 3 over denominator scriptbase log invisible function application open parentheses x plus 1 close parentheses end scriptbase presuperscript 9 end fraction end cell equals 5 row cell scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 3 plus fraction numerator 3 over denominator scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 3 squared end presuperscript end fraction end cell equals 5 row cell scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 3 plus fraction numerator 3 over denominator 1 half times scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 3 end fraction end cell equals 5 row cell y plus fraction numerator 3 over denominator 1 half times y end fraction end cell equals 5 row cell y plus 6 over y end cell equals 5 row cell y squared plus 6 end cell equals cell 5 y end cell row cell y squared minus 5 y plus 6 end cell equals 0 row cell left parenthesis y minus 2 right parenthesis left parenthesis y minus 3 right parenthesis end cell equals 0 row cell y minus 2 end cell equals cell 0 blank atau blank y minus 3 equals 0 end cell row y equals cell 2 blank atau blank y equals 3 end cell end table

Didapatkan hasil y subscript 1 equals 2 dan y subscript 2 equals 3, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell y subscript 1 end cell equals 2 row cell scriptbase log space invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 3 end cell equals 2 row cell 3 squared end cell equals cell x plus 1 end cell row 9 equals cell x plus 1 end cell row cell 9 minus 1 end cell equals x row 8 equals x end table

table attributes columnalign right center left columnspacing 0px end attributes row cell y subscript 2 end cell equals 3 row cell scriptbase log space invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 3 end cell equals 3 row cell 3 cubed end cell equals cell x plus 1 end cell row 27 equals cell x plus 1 end cell row cell 27 minus 1 end cell equals x row 26 equals x end table

Didapatkan hasil x subscript 1 equals 8 dan x subscript 2 equals 26, sehingga hasil dari x subscript 1 plus x subscript 2 adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 plus x subscript 2 end cell equals cell 8 plus 26 end cell row blank equals 34 end table

Jadi, dapat disimpulkan bahwa persamaan logaritma: scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 3 plus fraction numerator 3 over denominator scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 9 end fraction equals 5 mempunyai akar-akar x subscript 1 dan x subscript 2 maka x subscript 1 plus x subscript 2 equals 34.

Oleh karena itu, jawaban yang benar adalah A.

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

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

Nilai yang memenuhi persamaan:  adalah ...

Pembahasan Soal:

Ingat materi persamaan logaritma dengan basisnya adalah fungsi yang sama dan numerusnya adalah fungsi yang sama dan 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. Ingat juga sifat-sifat pada bentuk logaritma yaitu scriptbase log invisible function application a end scriptbase presuperscript a equals 1

Untuk menyelesaikan persamaan tersebut maka dilakkukan permisalan terlebih dahulu.

Misalkan log invisible function application x equals y sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses log invisible function application x close parentheses squared minus 4 open parentheses log invisible function application x close parentheses plus 3 end cell equals 0 row cell y squared minus 4 y plus 3 end cell equals 0 row cell left parenthesis y minus 1 right parenthesis left parenthesis y minus 3 right parenthesis end cell equals 0 row cell y minus 1 end cell equals cell 0 blank atau blank y minus 3 equals 0 end cell row y equals cell 1 blank atau blank y equals 3 end cell end table

Didapatkan hasil y subscript 1 equals 1 dan y subscript 2 equals 3, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell y subscript 1 end cell equals 1 row cell log invisible function application x end cell equals 1 row cell scriptbase log invisible function application x end scriptbase presuperscript 10 end cell equals 1 row cell 10 to the power of 1 end cell equals x row 10 equals x end table

table attributes columnalign right center left columnspacing 0px end attributes row cell y subscript 2 end cell equals 3 row cell log invisible function application x end cell equals 3 row cell scriptbase log invisible function application x end scriptbase presuperscript 10 end cell equals 3 row cell 10 cubed end cell equals x row 1000 equals x end table

Jadi, dapat disimpulkan bahwa nilai x yang memenuhi persamaan: left parenthesis log invisible function application x right parenthesis squared minus 4 open parentheses log invisible function application x close parentheses plus 3 equals 0 adalah x equals 10 dan x equals 1000.

Oleh karena itu, jawaban yang benar adalah E.

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