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Nilai x yang memenuhi persamaan 42x+1⋅34x=64 adalah...

Pertanyaan

Nilai x yang memenuhi persamaan 4 to the power of 2 x plus 1 end exponent times 3 to the power of 4 x end exponent equals 64 adalah...

  1. Error converting from MathML to accessible text.

  2. Error converting from MathML to accessible text.

  3. Error converting from MathML to accessible text.

  4. Error converting from MathML to accessible text.

  5. Error converting from MathML to accessible text.

Pembahasan Soal:

Ingat kembali:

a to the power of n equals stack stack a cross times a cross times a cross times... cross times a with underbrace below with a space s e b a n y a k space n below comma space a not equal to 0

open parentheses a to the power of m close parentheses to the power of n equals a to the power of m times n end exponent comma space a not equal to 0 

a to the power of b equals c rightwards arrow b equals log presuperscript a space c 

log presuperscript a space b to the power of n equals n log presuperscript a space b space comma space a greater than 0  

Sehingga diperoleh perhitungan:

table attributes columnalign right center left columnspacing 0px end attributes row cell 4 to the power of 2 x plus 1 end exponent times 3 to the power of 4 x end exponent end cell equals 64 row cell 4 to the power of 2 x end exponent times 4 times 3 to the power of 4 x end exponent end cell equals 64 row cell open parentheses 2 squared close parentheses to the power of 2 x end exponent times 3 to the power of 4 x end exponent end cell equals cell 64 over 4 end cell row cell 2 to the power of 4 x end exponent times 3 to the power of 4 x end exponent end cell equals 16 row cell open parentheses 2 times 3 close parentheses to the power of 4 x end exponent end cell equals cell 2 to the power of 4 end cell row cell 6 to the power of 4 x end exponent end cell equals cell 2 to the power of 4 end cell row cell 4 x end cell equals cell log presuperscript 6 space 2 to the power of 4 end cell row x equals cell 1 fourth space log presuperscript 6 space 2 to the power of 4 end cell row x equals cell space log presuperscript 6 space 2 to the power of 4 open parentheses 1 fourth close parentheses end exponent end cell row x equals cell log presuperscript 6 space 2 end cell end table 

Dengan demikian, Nilai x yang memenuhi persamaan tersebut adalah Error converting from MathML to accessible text.

Jadi, jawaban yang tepat adalah D

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

M. Nasrullah

Mahasiswa/Alumni Universitas Negeri Makassar

Terakhir diupdate 06 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Akar-akar dari  23logx+3logx3​=5  adalah x1​danx2​. Nilai x12​+x22​=...

Pembahasan Soal:

Ingat kembali:

a to the power of n equals stack stack a cross times a cross times a cross times... cross times a with underbrace below with a space s e b a n y a k space n below comma space a not equal to 0

log presuperscript a space a to the power of n equals n comma space a greater than 0

log presuperscript a space b to the power of n equals n log presuperscript a space b space comma space a space dan space b greater than 0

log presuperscript a space f open parentheses x close parentheses equals log presuperscript a space g open parentheses x close parentheses rightwards arrow f open parentheses x close parentheses equals g open parentheses x close parentheses 

open parentheses a to the power of m close parentheses to the power of n equals a to the power of m times n end exponent comma space a not equal to 0

Misalkan:

log presuperscript 3 space x equals a

Sehingga diperoleh perhitungan:

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 log presuperscript 3 space x plus fraction numerator 3 over denominator log presuperscript 3 space x end fraction end cell equals 5 row cell 2 log presuperscript 3 space x plus fraction numerator 3 over denominator log presuperscript 3 space x end fraction minus 5 end cell equals 0 row cell 2 a plus 3 over a minus 5 end cell equals cell 0 space open parentheses sama minus sama space dikali space straight a close parentheses end cell row cell a open parentheses 2 a close parentheses plus a open parentheses 3 over a close parentheses minus a open parentheses 5 close parentheses end cell equals 0 row cell 2 a squared plus 3 minus 5 a end cell equals 0 row cell 2 a squared minus 5 a plus 3 end cell equals cell 0 space open parentheses difaktorkan close parentheses end cell row cell open parentheses 2 a minus 3 close parentheses open parentheses a minus 1 close parentheses end cell equals 0 row blank blank cell a equals 3 over 2 space text atau  end text a equals 1 end cell end table 

Untuk a equals 1

table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript 3 space x end cell equals a row cell log presuperscript 3 space x end cell equals 1 row cell log presuperscript 3 space x end cell equals cell log presuperscript 3 space 3 end cell row cell x subscript 1 end cell equals 3 end table  

Untuk a equals 3 over 2

table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript 3 space x end cell equals a row cell log presuperscript 3 space x end cell equals cell 3 over 2 end cell row cell log presuperscript 3 space x end cell equals cell log presuperscript 3 space 3 to the power of 3 over 2 end exponent end cell row cell x subscript 2 end cell equals cell space 3 to the power of 3 over 2 end exponent end cell end table 

Sehingga,

table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 squared plus straight x subscript 2 squared end cell equals cell open parentheses 3 close parentheses squared plus open parentheses 3 to the power of 3 over 2 end exponent close parentheses squared end cell row blank equals cell 9 plus 3 to the power of 3 over 2 times 2 end exponent end cell row blank equals cell 9 plus 3 cubed end cell row blank equals cell 9 plus 27 end cell row blank equals 36 end table 

Dengan demikian, nilai x subscript 1 squared plus x subscript 2 squared   adalah 36

Jadi, jawaban yang tepat adalah B

0

Roboguru

Akar-akar dari  12(2logx​)+2logx1​=5 adalah x1​danx2​. Nilai x16​+x26​=...

Pembahasan Soal:

Ingat kembali:

a to the power of n equals stack stack a cross times a cross times a cross times... cross times a with underbrace below with a space s e b a n y a k space n below comma space a not equal to 0

log presuperscript a space a to the power of n equals n comma space a greater than 0

log presuperscript a space b to the power of n equals n log presuperscript a space b space comma space a space dan space b greater than 0

log presuperscript a space f open parentheses x close parentheses equals log presuperscript a space g open parentheses x close parentheses rightwards arrow f open parentheses x close parentheses equals g open parentheses x close parentheses 

open parentheses a to the power of m close parentheses to the power of n equals a to the power of m times n end exponent comma space a not equal to 0

Misalkan:

log presuperscript 2 space x equals a

Sehingga diperoleh perhitungan:

table attributes columnalign right center left columnspacing 0px end attributes row cell 12 open parentheses log presuperscript 2 space square root of x close parentheses plus fraction numerator 1 over denominator log presuperscript 2 space x end fraction end cell equals 5 row cell 12 open parentheses log presuperscript 2 space x to the power of 1 half end exponent close parentheses plus fraction numerator 1 over denominator log presuperscript 2 space x end fraction minus 5 end cell equals 0 row cell 12 times 1 half open parentheses log presuperscript 2 space x close parentheses plus fraction numerator 1 over denominator log presuperscript 2 space x end fraction minus 5 end cell equals 0 row cell 6 open parentheses log presuperscript 2 space x close parentheses plus fraction numerator 1 over denominator log presuperscript 2 space x end fraction minus 5 end cell equals 0 row cell 6 straight a plus 1 over straight a minus 5 end cell equals cell 0 space open parentheses sama minus sama space dikali space straight a close parentheses end cell row cell straight a open parentheses 6 straight a close parentheses plus straight a open parentheses 1 over straight a close parentheses minus straight a open parentheses 5 close parentheses end cell equals 0 row cell 6 straight a squared plus 1 minus 5 straight a end cell equals 0 row cell 6 straight a squared minus 5 straight a plus 1 end cell equals cell 0 space open parentheses difaktorkan close parentheses end cell row cell open parentheses 3 straight a minus 1 close parentheses open parentheses 2 straight a minus 1 close parentheses end cell equals 0 row blank blank cell straight a equals 1 third space atau space straight a equals 1 half end cell end table    

Untuk a equals 1 third 

table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript 2 space x end cell equals a row cell log presuperscript 2 space x end cell equals cell 1 third end cell row cell log presuperscript 2 space x end cell equals cell log presuperscript 2 space 2 to the power of 1 third end exponent end cell row cell x subscript 1 end cell equals cell 2 to the power of 1 third end exponent end cell end table  

Untuk a equals 1 half 

  table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript 2 space x end cell equals a row cell log presuperscript 2 space x end cell equals cell 1 half end cell row cell log presuperscript 2 space x end cell equals cell log presuperscript 2 space 2 to the power of 1 half end exponent end cell row cell x subscript 2 end cell equals cell 2 to the power of 1 half end exponent end cell end table 

Sehingga,

table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 to the power of 6 plus straight x subscript 2 to the power of 6 end cell equals cell open parentheses 2 to the power of 1 third end exponent close parentheses to the power of 6 plus open parentheses 2 to the power of 1 half end exponent close parentheses to the power of 6 end cell row blank equals cell 2 to the power of 1 third times 6 end exponent plus 2 to the power of 1 half times 6 end exponent end cell row blank equals cell 2 squared plus 2 cubed end cell row blank equals cell 4 plus 8 end cell row blank equals 12 end table  

Dengan demikian, nilai x subscript 1 to the power of 6 plus x subscript 2 to the power of 6  adalah 12

Jadi, jawaban yang tepat adalah A

0

Roboguru

Jumlah akar-akar persamaan: 2⋅3x+3x18​−14=0 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 x yang memenuhi persamaan x+1log(2x+5)=2⋅2log122log224−2log26​ 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

Misal persamaan logaritma: 3log(x+1)+9log(x+1)3​=5 mempunyai akar-akar x1​ dan x2​ maka x1​+x2​=…

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.

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Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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