jika²log 3 = x³log 5 = y[tex]_______________________[/tex]buktikan jika hasil ⁵log 12 = (x+2)/(xy).... Question from @Zdytx

henriyulianto

Terbukti benar bahwa:

${}^5\log12=\dfrac{x+2}{xy}$

Pembahasan

Logaritma

Diketahui:

${}^2\log3=x$
${}^3\log5=y$

Bentuk-bentuk lain yang ekuivalen berdasarkan sifat-sifat logaritma antara lain adalah:

$\begin{aligned}&{}^2\log3=x\\&\Rightarrow\frac{{}^5\log3}{{}^5\log2}=x\quad...(i)\\&\Rightarrow{}^5\log2=\frac{{}^5\log3}{x}\quad...(ii)\\&.........................................\\&{}^3\log5=y\\&\Rightarrow{}^5\log3=\frac{1}{y}\quad...(iii)\\&\!\!\stackrel{(ii)}{\implies}{}^5\log2=\frac{1}{xy}\quad...(iv)\\&.........................................\\\end{aligned}$

Kemudian, akan dibuktikan bahwa:

$\large\text{$\begin{aligned}\boxed{\ {}^5\log12=\frac{x+2}{xy}\ }\end{aligned}$}$

$\large\text{$\begin{aligned}&\textsf{Ruas kiri}={}^5\log12\\&{=\ }{}^5\log\left(2^2\cdot3\right)\\&{=\ }{}^5\log\left(2^2\right)+{}^5\log3\\&{=\ }2\cdot{}^5\log2+{}^5\log3\\&\ ...\ \textsf{substitusi dari $(iii)$ dan $(iv)$}\\&{=\ }2\cdot\frac{1}{xy}+\frac{1}{y}\\&{=\ }\frac{2}{xy}+\frac{x}{xy}\\&{=\ }\frac{x+2}{xy}=\textsf{Ruas kanan}\\&\blacksquare\quad\textsf{TERBUKTI}\end{aligned}$}$