Jika 7log 3=a dan 3log 4=b, tentukan nilai logaritma berikut. b. 12log 112

Pertanyaan

Jika ${}^7\log\;3=a$ dan ${}^3\log\;4=b$ , tentukan nilai logaritma berikut.

b. ${}^{12}\log\;112$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

nilai dari log presuperscript 12 space 112

adalah $fraction numerator 1 plus 2 a b over denominator a open parentheses b plus 1 close parentheses end fraction$ .

Pembahasan

Ingat sifat bentuk logaritma:

$table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript a space b end cell equals cell fraction numerator log presuperscript c space b over denominator log presuperscript c space a end fraction comma space c not equal to 1 end cell row cell log presuperscript a space a end cell equals 1 row cell log presuperscript a space b c end cell equals cell log presuperscript a space b plus log presuperscript a space c end cell row cell log presuperscript a space b to the power of m end cell equals cell m times log presuperscript a space b end cell end table$

Diketahui log presuperscript 7 space 3 equals a maka:

$table row cell log presuperscript 7 space 3 equals a end cell left right double arrow cell fraction numerator log space 3 over denominator log space 7 end fraction equals a end cell row blank left right double arrow cell fraction numerator log space 7 over denominator log space 3 end fraction equals 1 over a end cell row blank left right double arrow cell log presuperscript 3 space 7 equals 1 over a end cell end table$

sehingga:

$table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript 12 space 112 end cell equals cell fraction numerator log presuperscript 3 space 112 over denominator log presuperscript 3 space 12 end fraction end cell row blank equals cell fraction numerator log presuperscript 3 space open parentheses 7 cross times 16 close parentheses over denominator log presuperscript 3 space open parentheses 4 cross times 3 close parentheses end fraction end cell row blank equals cell fraction numerator log presuperscript 3 space 7 plus log presuperscript 3 space 16 over denominator log presuperscript 3 space 4 plus log presuperscript 3 space 3 end fraction end cell row blank equals cell fraction numerator log presuperscript 3 space 7 plus log presuperscript 3 space 4 squared over denominator log presuperscript 3 space 4 plus log presuperscript 3 space 3 end fraction end cell row blank equals cell fraction numerator log presuperscript 3 space 7 plus 2 times log presuperscript 3 space 4 over denominator log presuperscript 3 space 4 plus log presuperscript 3 space 3 end fraction end cell row blank equals cell fraction numerator begin display style 1 over a end style plus 2 times b over denominator b plus 1 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 1 plus 2 a b over denominator a end fraction end style over denominator b plus 1 end fraction end cell row blank equals cell fraction numerator 1 plus 2 a b over denominator a end fraction cross times fraction numerator 1 over denominator b plus 1 end fraction end cell row blank equals cell fraction numerator 1 plus 2 a b over denominator a open parentheses b plus 1 close parentheses end fraction end cell end table$

Dengan demikian nilai dari log presuperscript 12 space 112 adalah $fraction numerator 1 plus 2 a b over denominator a open parentheses b plus 1 close parentheses end fraction$ .