Question

If $2^x=3^y={12}^z$, then find the value of $\frac{z(x+2y)}{xy}$.

Answer 1

Let

$2^x=3^y={12}^z=k$

$\therefore 2^x=k$

$x\log 2=\log k$

$x=\frac{\log k}{\log 2}$

Similarly,

$y=\frac{\log k}{\log 3},z=\frac{\log k}{\log 12}$

Since,

$\Rightarrow \frac{z(x+2y)}{xy}$

$\Rightarrow \frac{\frac{\log k}{\log 12}(\frac{\log k}{\log 2}+\frac{2\log k}{\log 3})}{\frac{\log k}{\log 2}\times \frac{\log k}{\log 3}}$

$\Rightarrow \frac{\frac{1}{\log 12}(\frac{1}{\log 2}+\frac{2}{\log 3})}{\frac{1}{\log 2}\times \frac{1}{\log 3}}$

$\Rightarrow \frac{1}{\log 12}(\log 3+2\log 2)$

$\Rightarrow \frac{1}{\log 12}(\log 3+\log 4)(\because n\log m=\log m^n)$

$\Rightarrow \frac{1}{\log 12}\left(\log 12\right)(\because \log a+\log b=\log ab)$

$\Rightarrow 1$

Hence, this is the answer.