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Tentukan nilai-nilai x persamaan-persamaan logaritma berikut!   a.3log(x−2)−9log(4x−8)=0

Pertanyaan

Tentukan nilai-nilai x persamaan-persamaan logaritma berikut! 

 a.3log(x2)9log(4x8)=0 

Pembahasan Soal:

Ingat 

alogf(x)=alogg(x)f(x)=g(x) 

Perhatikan perhitungan berikut 

3log(x2)9log(4x8)3log(x2)=9log(4x8)3log(x2)=32log(4x8)3log(x2)=213log(4x8)3log(x2)=3log(4x8)21(x2)(x2)2x24x+4x28x+12(x6)(x2)x6x========0(4x8)21(4x8)4x8000ataux2=06x=2 

Uji syarat numerus (numerus>0

Untuk x=2 

x2(2)20>>>000tidakmemenuhi 

Untuk x=6 

x2(6)244x84(6)824816>>>>>>>000memenuhi0000memenuhi 

Jadi, nilai x yang memenuhi adalah x=6

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Lestari

Mahasiswa/Alumni Universitas Sebelas Maret

Terakhir diupdate 13 September 2021

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Pertanyaan yang serupa

Tentukan nilai-nilai x persamaan-persamaan logaritma berikut!  b.3log(2−3x)=9log(7x2−9x+3)

Pembahasan Soal:

Ingat 

alogf(x)=alogg(x)f(x)=g(x) 

Perhatikan perhitungan berikut 

3log(23x)=9log(7x29x+3)3log(23x)=32log(7x29x+3)3log(23x)3log(23x)=3log(7x29x+3)2123x7x26x1(7x+1)(x1)7x+17xx=======213log(7x29x+3)7x29x+3000ataux1=01x=171 

Uji syarat numerus (numerus>0

Untuk x=71 

23x23(71)2+737177x29x+37(71)29(71)+349217>>>>>>>0000memenuhi000memenuhi 

Untuk x=1 

23x23(1)1>>>000tidakmemenuhi 

Jadi, nilai x yang memenuhi adalah x=71.  

0

Roboguru

Nilai x yang memenuhi persamaan   adalah...

Pembahasan Soal:

table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript x space open parentheses x plus 2 close parentheses minus 3 times log presuperscript x space 2 plus 1 end cell equals 0 row cell log presuperscript x space open parentheses x plus 2 close parentheses minus log presuperscript x space 2 cubed plus 1 end cell equals 0 row cell log presuperscript x space fraction numerator x plus 2 over denominator 2 cubed end fraction plus 1 end cell equals 0 row cell log presuperscript x space fraction numerator x plus 2 over denominator 8 end fraction end cell equals cell negative 1 end cell row cell fraction numerator x plus 2 over denominator 8 end fraction end cell equals cell x to the power of negative 1 end exponent end cell row cell fraction numerator x plus 2 over denominator 8 end fraction end cell equals cell 1 over x end cell row cell x open parentheses x plus 2 close parentheses end cell equals 8 row cell x squared plus 2 x end cell equals 8 row cell x squared plus 2 x minus 8 end cell equals 0 row cell open parentheses x minus 2 close parentheses open parentheses x plus 4 close parentheses end cell equals 0 row cell x minus 2 end cell equals 0 row x equals 2 row cell x plus 4 end cell equals 0 row x equals cell negative 4 end cell end table 

Jadi, nilai x yang memenuhi adalah x equals 2 space dan space x equals negative 4.

Oleh karena itu, jawaban yang benar adalah B.

0

Roboguru

Jika , tentukan nilai  yang memenuhi persamaan tersebut.

Pembahasan Soal:

Ingat sifat-sifat logaritma:

table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript a space b minus log presuperscript a space c end cell equals cell log presuperscript a space b over c end cell row cell log presuperscript a space a to the power of n end cell equals n row cell fraction numerator 1 over denominator log presuperscript a space b end fraction end cell equals cell log presuperscript b space a end cell 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 log presuperscript a space b end cell end table  

Ingat ada persamaan logaritma berlaku jika log presuperscript h open parentheses x close parentheses end presuperscript space f open parentheses x close parentheses equals log presuperscript h open parentheses x close parentheses end presuperscript space g open parentheses x close parentheses comma space a greater than 0 dan a not equal to 1 maka f open parentheses x close parentheses equals g open parentheses x close parentheses dengan syarat h open parentheses x close parentheses greater than 0 comma space h open parentheses x close parentheses not equal to 1f open parentheses x close parentheses greater than 0 dan g open parentheses x close parentheses greater than 0

Diketahui fraction numerator 1 over denominator log presuperscript 16 space open parentheses x minus 2 close parentheses minus log presuperscript 16 space open parentheses x squared minus 4 x plus 4 close parentheses end fraction equals negative 2 maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator log presuperscript 16 space open parentheses x minus 2 close parentheses minus log presuperscript 16 open parentheses x squared minus 4 x plus 4 close parentheses end fraction end cell equals cell negative 2 end cell row cell fraction numerator 1 over denominator log presuperscript 16 space begin display style fraction numerator x minus 2 over denominator x squared minus 4 x plus 4 end fraction end style end fraction end cell equals cell negative 2 end cell row cell fraction numerator 1 over denominator log presuperscript 16 space begin display style fraction numerator x minus 2 over denominator open parentheses x minus 2 close parentheses open parentheses x minus 2 close parentheses end fraction end style end fraction end cell equals cell negative 2 end cell row cell fraction numerator 1 over denominator log presuperscript 16 space begin display style fraction numerator 1 over denominator x minus 2 end fraction end style end fraction end cell equals cell negative 2 end cell row cell fraction numerator 1 over denominator log presuperscript 16 space open parentheses x minus 2 close parentheses to the power of negative 1 end exponent end fraction end cell equals cell negative 2 end cell row cell log presuperscript open parentheses x minus 2 close parentheses to the power of negative 1 end exponent end presuperscript space 16 end cell equals cell negative 2 end cell row cell negative log presuperscript open parentheses x minus 2 close parentheses end presuperscript space 16 end cell equals cell negative 2 end cell row cell log presuperscript open parentheses x minus 2 close parentheses end presuperscript space 16 space end cell equals 2 row cell log presuperscript open parentheses x minus 2 close parentheses end presuperscript space 16 end cell equals cell log presuperscript open parentheses x minus 2 close parentheses end presuperscript space open parentheses x minus 2 close parentheses squared end cell row 16 equals cell open parentheses x minus 2 close parentheses squared end cell row 16 equals cell x squared minus 4 x plus 4 end cell row cell x squared minus 4 x plus 4 minus 16 end cell equals 0 row cell x squared minus 4 x minus 12 end cell equals 0 row cell open parentheses x minus 6 close parentheses open parentheses x plus 2 close parentheses end cell equals 0 end table  

diperoleh nilai yang memenuhi x equals 6 atau x equals negative 2  

Syarat numerus 1)

table attributes columnalign right center left columnspacing 0px end attributes row cell x minus 2 end cell greater than 0 row x greater than 2 end table   

syarat numerus 2)

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared minus 4 x plus 4 end cell greater than 0 row cell open parentheses x minus 2 close parentheses open parentheses x minus 2 close parentheses end cell greater than 0 row x greater than 2 end table  

Dari syarat numerus mengharuskan x greater than 2. Dengan demikian nilai x yang memenuhi adalah x equals 6.

 

0

Roboguru

Nilai yang memenuhi persamaan:  adalah ...

Pembahasan Soal:

Ingat sifat pada persamaan bentuk logaritma yaitu scriptbase log invisible function application f left parenthesis x right parenthesis end scriptbase presuperscript a equals scriptbase log invisible function application g left parenthesis x right parenthesis end scriptbase presuperscript b rightwards arrow f open parentheses x close parentheses equals g left parenthesis x right parenthesis dan 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 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

Ingat syarat numerus pada bentuk logaritma scriptbase log invisible function application b end scriptbase presuperscript a yaitu b greater than 0. Sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell log invisible function application open parentheses x squared minus 24 close parentheses end cell equals cell log invisible function application x plus 7 log invisible function application 16 over 15 plus 5 log invisible function application 25 over 24 plus 3 log invisible function application 81 over 80 end cell row cell log invisible function application left parenthesis x squared minus 24 right parenthesis end cell equals cell log invisible function application x plus log invisible function application open parentheses 16 over 15 close parentheses to the power of 7 plus log invisible function application open parentheses 25 over 24 close parentheses to the power of 5 plus log invisible function application open parentheses 81 over 80 close parentheses cubed end cell row cell log invisible function application left parenthesis x squared minus 24 right parenthesis end cell equals cell log invisible function application open parentheses x times open parentheses 16 over 15 close parentheses to the power of 7 times open parentheses 25 over 24 close parentheses to the power of 5 times open parentheses 81 over 80 close parentheses cubed close parentheses end cell row cell log invisible function application left parenthesis x squared minus 24 right parenthesis end cell equals cell log invisible function application open parentheses x times open parentheses fraction numerator 2 to the power of 4 over denominator 3 times 5 end fraction close parentheses to the power of 7 times open parentheses fraction numerator 5 squared over denominator 8 times 3 end fraction close parentheses to the power of 5 times open parentheses fraction numerator 3 to the power of 4 over denominator 16 times 5 end fraction close parentheses cubed close parentheses end cell row cell log invisible function application left parenthesis x squared minus 24 right parenthesis end cell equals cell log invisible function application open parentheses x times open parentheses fraction numerator 2 to the power of 4 over denominator 3 times 5 end fraction close parentheses to the power of 7 times open parentheses fraction numerator 5 squared over denominator 2 cubed times 3 end fraction close parentheses to the power of 5 times open parentheses fraction numerator 3 to the power of 4 over denominator 2 to the power of 4 times 5 end fraction close parentheses cubed close parentheses end cell row cell log invisible function application left parenthesis x squared minus 24 right parenthesis end cell equals cell log invisible function application open parentheses x times fraction numerator 2 to the power of 28 over denominator 3 to the power of 7 times 5 to the power of 7 end fraction times fraction numerator 5 to the power of 10 over denominator 2 to the power of 15 times 3 to the power of 5 end fraction times fraction numerator 3 to the power of 12 over denominator 2 to the power of 12 times 5 cubed end fraction close parentheses end cell row cell log invisible function application left parenthesis x squared minus 24 right parenthesis end cell equals cell log invisible function application open parentheses x times fraction numerator 2 to the power of 28 times 3 to the power of 12 times 5 to the power of 10 over denominator 2 to the power of 27 times 3 to the power of 12 times 5 to the power of 10 end fraction close parentheses end cell row cell log invisible function application left parenthesis x squared minus 24 right parenthesis end cell equals cell log invisible function application open parentheses x times 2 to the power of 28 over 2 to the power of 27 close parentheses end cell row cell log invisible function application left parenthesis x squared minus 24 right parenthesis end cell equals cell log invisible function application open parentheses x times 2 close parentheses end cell row cell log invisible function application left parenthesis x squared minus 24 right parenthesis end cell equals cell log invisible function application 2 x end cell row cell x squared minus 24 end cell equals cell 2 x end cell row cell x squared minus 2 x minus 24 end cell equals 0 row cell left parenthesis x minus 6 right parenthesis left parenthesis x plus 4 right parenthesis end cell equals 0 row cell x minus 6 end cell equals cell 0 blank atau blank x plus 4 equals 0 end cell row x equals cell 6 blank atau blank x equals negative 4 end cell end table

Didapatkan nilai x yaitu x equals 6 dan x equals negative 4. Karena numerus pada bentuk logaritma harus positif maka nilai x yang memenuhi adalah x equals 6, karena untuk x equals negative 4 maka hasil dari x squared minus 24 dan 2 x akan bernilai negatif. 

Jadi, dapat disimpulkan bahwa nilai x yang memenuhi persamaan: adalah log invisible function application left parenthesis x squared minus 24 right parenthesis equals log invisible function application x plus 7 log invisible function application 16 over 15 plus 5 log invisible function application 25 over 24 plus 3 log invisible function application 81 over 80 adalah x equals 6.

Oleh karena itu, jawaban yang benar adalah C.

0

Roboguru

Akar-akar dari  adalah  dan  maka

Pembahasan Soal:

Ingat sifat pada persamaan bentuk logaritma yaitu scriptbase log invisible function application f left parenthesis x right parenthesis end scriptbase presuperscript a equals scriptbase log invisible function application g left parenthesis x right parenthesis end scriptbase presuperscript b rightwards arrow f open parentheses x close parentheses equals g left parenthesis x right parenthesis dan sifat pada bentuk logaritma yaitu

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

Ingat 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 dan syarat numerus pada logaritma scriptbase log invisible function application b end scriptbase presuperscript a yaitu b greater than 0. Sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 3 end cell equals cell 1 plus scriptbase log invisible function application left parenthesis x minus 1 right parenthesis end scriptbase presuperscript 9 end cell row cell scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 3 end cell equals cell scriptbase log invisible function application 3 end scriptbase presuperscript 3 plus scriptbase log invisible function application left parenthesis x minus 1 right parenthesis end scriptbase presuperscript 3 squared end presuperscript end cell row cell scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 3 end cell equals cell scriptbase log invisible function application 3 end scriptbase presuperscript 3 plus 1 half scriptbase log invisible function application left parenthesis x minus 1 right parenthesis end scriptbase presuperscript 3 end cell row cell scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 3 end cell equals cell scriptbase log invisible function application 3 end scriptbase presuperscript 3 plus scriptbase log invisible function application open parentheses left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent close parentheses end scriptbase presuperscript 3 end cell row cell scriptbase log invisible function application left parenthesis x plus 1 right parenthesis end scriptbase presuperscript 3 end cell equals cell scriptbase log invisible function application open parentheses 3 times left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent close parentheses end scriptbase presuperscript 3 end cell row cell x plus 1 end cell equals cell 3 times left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent end cell row cell open parentheses x plus 1 close parentheses squared end cell equals cell open parentheses 3 times left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent close parentheses squared end cell row cell x squared plus 2 x plus 1 end cell equals cell 3 squared times left parenthesis x minus 1 right parenthesis end cell row cell x squared plus 2 x plus 1 end cell equals cell 9 times left parenthesis x minus 1 right parenthesis end cell row cell x squared plus 2 x plus 1 end cell equals cell 9 x minus 9 end cell row cell x squared plus 2 x minus 9 x plus 1 plus 9 end cell equals 0 row cell x squared minus 7 x plus 10 end cell equals 0 row cell left parenthesis x minus 2 right parenthesis left parenthesis x minus 5 right parenthesis end cell equals 0 row cell x minus 2 end cell equals cell 0 blank atau blank x minus 5 equals 0 end cell row x equals cell 2 blank atau blank x equals 5 end cell end table

Didapatkan akar-akar dari persamaan tersebut yaitu x subscript 1 equals 2 dan x equals 5. Maka nilai dari x subscript 1 plus x subscript 2 sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 plus x subscript 2 end cell equals cell 2 plus 5 end cell row blank equals 7 end table

Jadi, dapat disimpulkan bahwa nilai dari x subscript 1 plus x subscript 2 adalah 7.

Oleh karena itu, jawaban yang benar adalah D.

0

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