Answer
$\frac{\log_{a} (x+h)-\log_{a} x}{h}=\log_{a} (1+\frac{h}{x})^{\frac{1}{h}}$
Work Step by Step
$f(x)=\log_{a} {x}$
We want to prove:
$ \frac{f(x+h)-f(x)}{h}=\log(1+\frac{h}{x})^{\frac{1}{h}}, h\ne 0$
We have:
$\frac{\log_{a} (x+h)-\log_{a} x}{h}\\=\frac{\log (\frac{x+h}{x})}{h}\\=\frac{1}{h}\log(x+\frac{h}{x})\\=\log_{a} (1+\frac{h}{x})^{\frac{1}{h}}$
