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Tentukan nilai x yang memenuhi persamaan logaritma berikut! log(x−3)+log(x−2)=log(2x+4)

Pertanyaan

Tentukan nilai x yang memenuhi persamaan logaritma berikut!

log(x3)+log(x2)=log(2x+4)        

Pembahasan Soal:

Ingat 

alogf(x)=alogg(x)f(x)=g(x) 

Perhatikan perhitungan berikut 

log(x3)+log(x2)log[(x3)(x2)]log(x25x+6)x25x+6x27x+2x27x+449+2449(x27x+449)+48449(x27)2441(x27)2x27xxx=============log(2x+4)log(2x+4)log(2x+4)2x+40000441±44127±44127+441=455atau27441=427 

Karena numerus harus lebih besar dari 0, maka x=427 tidak memenuhi. 

Dengan demikian, nilai x yang memenuhi adalah x=455

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

P. Nur

Terakhir diupdate 17 September 2021

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Pertanyaan yang serupa

Tentukan penyelesaian persamaan logaritma berikut: a.

Pembahasan Soal:

Ingat sifat logaritma berikut!

log presuperscript a space b plus log presuperscript a space c equals log presuperscript a space b c 

table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript 4 space open parentheses x minus 4 close parentheses plus log presuperscript 4 space open parentheses x minus 2 close parentheses end cell equals cell 3 over 2 end cell row cell log presuperscript 4 space open parentheses x minus 4 close parentheses open parentheses x minus 2 close parentheses end cell equals cell 3 over 2 end cell row cell log presuperscript 4 space x squared minus 6 x plus 8 end cell equals cell 3 over 2 end cell end table 

Gunakan definisi logaritma untuk mendapatkan nilai x

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

table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript 4 space x squared minus 6 x plus 8 end cell equals cell 3 over 2 end cell row cell space x squared minus 6 x plus 8 end cell equals cell 4 to the power of 3 over 2 end exponent end cell row cell space x squared minus 6 x plus 8 end cell equals cell square root of 4 cubed end root end cell row cell space x squared minus 6 x plus 8 end cell equals cell square root of 64 end cell row cell space x squared minus 6 x plus 8 end cell equals 8 row cell space x squared minus 6 x plus 8 minus 8 end cell equals 0 row cell space x squared minus 6 x end cell equals 0 row cell x open parentheses x minus 6 close parentheses end cell equals 0 row x equals 0 row cell x minus 6 end cell equals 0 row x equals 6 end table 

Jadi, penyelesaian persamaan logaritma di atas adalah x equals 0 atau x equals 6.

Roboguru

Tentukan penyelesaian persamaan logaritma berikut: b.

Pembahasan Soal:

Ingat sifat logaritma berikut!

log presuperscript a space b plus log presuperscript a space c equals log presuperscript a space b c 

 table attributes columnalign right center left columnspacing 0px end attributes row cell log space open parentheses 2 x minus 3 close parentheses plus log space open parentheses x minus 4 close parentheses end cell equals cell log space open parentheses x minus 3 close parentheses plus log space x end cell row cell log space open parentheses 2 x minus 3 close parentheses open parentheses x minus 4 close parentheses end cell equals cell log space open parentheses x minus 3 close parentheses x end cell row cell open parentheses 2 x minus 3 close parentheses open parentheses x minus 4 close parentheses end cell equals cell open parentheses x minus 3 close parentheses x end cell row cell 2 x squared minus 11 x plus 12 end cell equals cell x squared minus 3 x end cell row cell x squared minus 8 x plus 12 end cell equals 0 row cell open parentheses x minus 2 close parentheses open parentheses x minus 6 close parentheses end cell equals 0 row cell x minus 2 end cell equals 0 row x equals 2 row cell x minus 6 end cell equals 0 row x equals 6 end table 

Jadi, penyelesaian persamaan logaritma di atas adalah x equals 2 atau x equals 6.

Roboguru

Himpunan penyelesaian persamaan  adalah...

Pembahasan Soal:

table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript 6 space open parentheses x squared minus 2 x minus 3 close parentheses end cell equals cell log presuperscript 6 space open parentheses x minus 2 close parentheses plus log presuperscript 6 space open parentheses x plus 3 close parentheses end cell row cell log presuperscript 6 space open parentheses x squared minus 2 x minus 3 close parentheses end cell equals cell log presuperscript 6 space open parentheses x minus 2 close parentheses open parentheses x plus 3 close parentheses end cell row cell x squared minus 2 x minus 3 end cell equals cell open parentheses x minus 2 close parentheses open parentheses x plus 3 close parentheses end cell row cell x squared minus 2 x minus 3 end cell equals cell x squared plus x minus 6 end cell row cell negative 3 x end cell equals cell negative 3 end cell row x equals 1 end table 

Jadi, himpunan penyelesaian persamaan logaritma di atas adalah open curly brackets 1 close curly brackets.

Oleh karena itu, jawaban yang benar adalah B.

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.

Roboguru

Tentukan penyelesaian persamaan berikut. b.

Pembahasan Soal:

Ingat sifat bentuk logaritma:

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

Ingat ada persamaan logaritma berlaku jika log presuperscript a space f open parentheses x close parentheses equals log presuperscript a 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 f open parentheses x close parentheses greater than 0 dan g open parentheses x close parentheses greater than 0

Diketahui log presuperscript 2 space open parentheses x plus 7 close parentheses plus log presuperscript 2 space open parentheses x plus 6 close parentheses plus log presuperscript begin inline style 1 half end style end presuperscript space open parentheses x plus 10 close parentheses equals 0 maka:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript 2 space open parentheses x plus 7 close parentheses plus log presuperscript 2 space open parentheses x plus 6 close parentheses plus log presuperscript begin inline style 1 half end style end presuperscript space open parentheses x plus 10 close parentheses end cell equals 0 row cell log presuperscript 2 space open parentheses x plus 7 close parentheses plus log presuperscript 2 space open parentheses x plus 6 close parentheses plus log presuperscript begin inline style 2 to the power of negative 1 end exponent end style end presuperscript space open parentheses x plus 10 close parentheses end cell equals 0 row cell log presuperscript 2 space open parentheses x plus 7 close parentheses plus log presuperscript 2 space open parentheses x plus 6 close parentheses plus open parentheses fraction numerator 1 over denominator negative 1 end fraction close parentheses log presuperscript begin inline style 2 end style end presuperscript space open parentheses x plus 10 close parentheses end cell equals 0 row cell log presuperscript 2 space open parentheses x plus 7 close parentheses plus log presuperscript 2 space open parentheses x plus 6 close parentheses minus log presuperscript begin inline style 2 end style end presuperscript space open parentheses x plus 10 close parentheses end cell equals 0 row cell log presuperscript 2 space open parentheses x plus 7 close parentheses plus log presuperscript 2 space open parentheses x plus 6 close parentheses end cell equals cell log presuperscript begin inline style 2 end style end presuperscript space open parentheses x plus 10 close parentheses end cell row cell log presuperscript 2 space open parentheses x plus 7 close parentheses open parentheses x plus 6 close parentheses end cell equals cell log presuperscript 2 space open parentheses x plus 10 close parentheses end cell row cell log presuperscript 2 space open parentheses x squared plus 13 x plus 42 close parentheses end cell equals cell log presuperscript 2 space open parentheses x plus 10 close parentheses end cell row cell x squared plus 13 x plus 42 end cell equals cell x plus 10 end cell row cell x squared plus 13 x minus x plus 42 minus 10 end cell equals 0 row cell x squared plus 12 x plus 32 end cell equals 0 row cell open parentheses x plus 8 close parentheses open parentheses x plus 4 close parentheses end cell equals 0 end table end style   

diperoleh nilai yang memenuhi x equals negative 8 atau x equals negative 4 

Syarat numerus 1)

table attributes columnalign right center left columnspacing 0px end attributes row cell x plus 7 end cell greater than 0 row x greater than cell negative 7 end cell end table

Syarat numerus 3)

table attributes columnalign right center left columnspacing 0px end attributes row cell x plus 6 end cell greater than 0 row x greater than cell negative 6 end cell end table 

Syarat numerus 2)

table attributes columnalign right center left columnspacing 0px end attributes row cell x plus 10 end cell greater than 0 row x greater than cell negative 10 end cell end table  

Dari syarat numerus mengharuskan x greater than negative 6 maka nilai yang memenuhi adalah x equals negative 4.

Dengan demikian himpunan penyelesaiany adalah open curly brackets negative 4 close curly brackets.

 

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