Question

Prove that the function $f\left(x\right)=lo{g}_{a}x$ is increasing on $(0,\infty )$ if $a>1$ abd decreasing on (0,x) if 0<a<1.

Answer 1

$f\left(x\right)=lo{g}_a^x$

$f^{\prime }\left(x\right)=\frac{1}{xlo{g}_e^a}$

Increasing if $f\left(x\right)<0$

$\frac{1}{xlo{g}_e^a}<0$ if $a>1$

Decreasing if $f^{\prime }\left(x\right)>0$

$\frac{1}{xlo{g}_e^a}>0$ if $a<1$.