Show that if $3^x+4^y=12^z \Rightarrow z=\frac{xy}{x+y}$ Show that if $3^x+4^y=12^z \Rightarrow z=\frac{xy}{x+y}$
$\textbf{My Attempt}$
$$3^x+4^y=12^z \Rightarrow z=\frac{xy}{x+y}$$
$$\Rightarrow \log(3^x4^y)=\log(12^z)\Rightarrow \log \big(\frac{3^x4^y}{12^z}\big)=0\Rightarrow \frac{3^x4^y}{12^z}=1 \Rightarrow 3^x4^y=12^z$$
$$\Rightarrow 3^x4^y=3^x+4^y$$
How do you find what $z$ equals from this? Any help would be appreciated.

It's wrong.
Try $x=z=1$ and $y=\log_23.$
