Property of Inequality for Logarithms

The statements will be proved one at a time.

$b>1$

If $b$ is greater than $1,$ the logarithmic function $f(x)={\log }_{b}x$ is increasing over its entire domain.

In the graph, it can be seen that $f(x_1)\ge f(x_2)$ if and only if $x_1\ge x_2.$ Considering the definition of $f(x),$ the statement is proven.

$b<1$

If $b$ is greater than $0$ and less than $1,$ then $f(x)={\log }_{b}x$ is decreasing over its entire domain.

In the graph, it can be seen that $f(x_1)\ge f(x_2)$ if and only if $x_1\le x_2.$ Considering the definition of $f(x),$ the statement is proven.