Nilai dari (9log4)(9log12)/((9log8)(3log60)-(9log8)(3log5))

1/3

Pembahasan

Kita perlu menyederhanakan ekspresi matematika berikut: (9log4)(9log12)/((9log8)(3log60)-(9log8)(3log5)).
Mari kita gunakan sifat logaritma: log_b(a^c) = c * log_b(a) dan log_b(a) * log_c(d) = log_c(d) * log_b(a).
Ekspresi = (9log4)(9log12) / (9log8)(3log60 - 3log5)
Menggunakan sifat logaritma log_b(a) - log_b(c) = log_b(a/c):
3log60 - 3log5 = 3log(60/5) = 3log12.
Ekspresi = (9log4)(9log12) / ((9log8)(3log12)).
Menggunakan sifat logaritma log_b(a) = log_c(a) / log_c(b) untuk mengubah basis:
9log4 = log(4)/log(9) = log(2^2)/log(3^2) = 2log(2)/2log(3) = log(2)/log(3) = 3log2.
9log12 = log(12)/log(9) = log(2^2*3)/log(3^2) = (2log2+log3)/2log3.
9log8 = log(8)/log(9) = log(2^3)/log(3^2) = 3log2/2log3.
3log12 = log(12)/log(3) = log(2^2*3)/log3 = (2log2+log3)/log3.

Substitusi kembali ke ekspresi:
Ekspresi = ( (log2/log3) * ((2log2+log3)/2log3) ) / ( (3log2/2log3) * ((2log2+log3)/log3) )
Ekspresi = ( (log2 * (2log2+log3)) / (2 * (log3)^2) ) / ( (3log2 * (2log2+log3)) / (2 * (log3)^2) )
Ekspresi = (log2 * (2log2+log3)) / (3log2 * (2log2+log3))
Ekspresi = log2 / (3log2) = 1/3.

Jawaban singkatnya adalah 1/3.