Question

Show that $f\left(x\right)={\log }_a{x}^{}$, $0<a<1$ is a decreasing function for all $x>0$.

Answer 1

$f\left(x\right)={\log }_a\left(x\right)$ $0<a<1$

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

Since $0<a<1$ therefore $\log a<0$

Now $x>0\Rightarrow \frac{1}{x}>0\Rightarrow \frac{1}{x\log a}<0\Rightarrow f^{\prime }\left(x\right)<0$

so $f\left(x\right)$ is decreasing $\forall x>0$