The range of ${\log }_b\left(x\right)$ is all real numbers.

The domain of ${\log }_b\left(x\right)$ is all positive real numbers.

For $b\&gt;1$, ${\log }_b\left(x\right)\&gt;0$ for $x\&gt;1$ and ${\log }_b\left(x\right)<0$ for $0<x<1$; for $0<b<1$ the inequalities reverse.

${\log }_b\left(1\right)\&equals;0$

${\log }_b\left(x\cdot y\right)\&equals;{\log }_b\left(x\right)\&plus;{\log }_b\left(y\right)$

${\log }_b\left(\frac{x}{y}\right)\&equals;{\log }_b\left(x\right)-{\log }_b\left(y\right)$

${\log }_b\left(\frac{1}{x}\right)\&equals;-{\log }_b\left(x\right)$

${\log }_b\left(x^r\right)\&equals;r {\log }_b\left(x\right)$

If $x\&gt;y$ and $b\&gt;1$ then ${\log }_b\left(x\right)\&gt;{\log }_b\left(y\right)$. If $x\&gt;y$ and $0<b<1$ then ${\log }_b\left(x\right)<{\log }_b\left(y\right)$. That is, ${\log }_b\left(x\right)$ is an increasing function if $b\&gt;1$ and a decreasing function if $0<b<1$.

$x\&equals;{\log }_b\left(y\right)$ exactly when $y\&equals;b^x$. That is, the logarithmic and exponential functions with the same base are inverses of each other. In particular, ${\log }_b\left(b^x\right)\&equals;x\&equals;b^{{\log }_b\left(x\right)}$.