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Nilai x yang memenuhi persamaan 2log2log(2x+1+3)=1+2logx adalah ...

Pertanyaan

Nilai begin mathsize 14px style x end style yang memenuhi persamaan begin mathsize 14px style scriptbase log scriptbase log open parentheses 2 to the power of x plus 1 end exponent plus 3 close parentheses equals 1 plus log presuperscript 2 space x end scriptbase presuperscript 2 end scriptbase presuperscript 2 end style adalah ... 

  1. begin mathsize 14px style log space 2 over 3 end style 

  2. begin mathsize 14px style scriptbase log space 3 end scriptbase presuperscript 2 end style 

  3. begin mathsize 14px style scriptbase log space 2 end scriptbase presuperscript 3 end style 

  4. begin mathsize 14px style negative 1 end style atau begin mathsize 14px style 3 end style 

  5. begin mathsize 14px style 8 end style atau begin mathsize 14px style 1 half end style 

Pembahasan:

Diketahui:

Persamaan begin mathsize 14px style scriptbase log scriptbase log open parentheses 2 to the power of x plus 1 end exponent plus 3 close parentheses equals 1 plus log presuperscript 2 space x end scriptbase presuperscript 2 end scriptbase presuperscript 2 end style.

Ingatlah sifat-sifat logaritma:

begin mathsize 14px style table attributes columnalign left center left end attributes row cell scriptbase log space a end scriptbase presuperscript a end cell equals 1 row cell scriptbase log space b c end scriptbase presuperscript a end cell equals cell scriptbase log space b plus scriptbase log space c end scriptbase presuperscript a end scriptbase presuperscript a end cell row cell log presuperscript a space a to the power of n end cell equals n end table end style

Kemudian diperoleh persamaan logaritmanya, yaitu:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell scriptbase log log presuperscript 2 open parentheses 2 to the power of x plus 1 end exponent plus 3 close parentheses end scriptbase presuperscript 2 end cell equals cell 1 plus scriptbase log space x end scriptbase presuperscript 1 end cell row cell scriptbase log log presuperscript 2 open parentheses 2 to the power of x plus 1 end exponent plus 3 close parentheses end scriptbase presuperscript 2 end cell equals cell scriptbase log space 2 end scriptbase presuperscript 2 plus scriptbase log space x end scriptbase presuperscript 1 end cell row cell down diagonal strike log presuperscript 2 end strike scriptbase log open parentheses 2 to the power of x plus 1 end exponent plus 3 close parentheses end scriptbase presuperscript 2 end cell equals cell down diagonal strike log presuperscript 2 end strike space 2 x end cell row cell scriptbase log open parentheses 2 to the power of x plus 1 end exponent plus 3 close parentheses end scriptbase presuperscript 2 end cell equals cell 2 x end cell row cell down diagonal strike log presuperscript 2 end strike open parentheses 2 to the power of x plus 1 end exponent plus 3 close parentheses end cell equals cell down diagonal strike log presuperscript 2 end strike space 2 to the power of 2 x end exponent end cell row cell 2 to the power of x plus 1 end exponent plus 3 end cell equals cell 2 to the power of 2 x end exponent end cell row cell 2 times 2 to the power of x plus 3 end cell equals cell 2 to the power of 2 x end exponent end cell end table end style

Misal: begin mathsize 14px style 2 x equals a end style, sehingga diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 times 2 to the power of x plus 3 end cell equals cell 2 to the power of 2 x end exponent end cell row cell 2 to the power of 2 x end exponent minus 4 to the power of x minus 3 end cell equals 0 row cell open parentheses 2 to the power of x close parentheses squared minus 2 open parentheses 2 to the power of x close parentheses minus 3 end cell equals 0 row cell a squared minus 2 a minus 3 end cell equals 0 row cell open parentheses a minus 3 close parentheses open parentheses a plus 1 close parentheses end cell equals 0 end table end style

Dengan,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell a minus 3 end cell equals 0 row a equals 3 row blank blank blank row cell a plus 1 end cell equals 0 row a equals cell negative 1 end cell end table end style

Karena begin mathsize 14px style negative 1 end style tidak memenuhi bentuk logaritma, maka nilai yang dipakai adalah begin mathsize 14px style a equals 3 end style.

Dimisalkan begin mathsize 14px style 2 x equals a end style, maka:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 to the power of x end cell equals a row cell 2 to the power of x end cell equals 3 row x equals cell scriptbase log space 3 end scriptbase presuperscript 2 end cell end table end style

Nilai begin mathsize 14px style x end style yang memenuhi persamaan begin mathsize 14px style scriptbase log scriptbase log open parentheses 2 to the power of x plus 1 end exponent plus 3 close parentheses equals 1 plus log presuperscript 2 space x end scriptbase presuperscript 2 end scriptbase presuperscript 2 end style adalah begin mathsize 14px style scriptbase log space 3 end scriptbase presuperscript 2 end style.

Jadi, jawaban yang tepat adalah B.

Jawaban terverifikasi

Dijawab oleh:

S. Ayu

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Terakhir diupdate 06 Oktober 2021

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