Take a look at the applet: Logarithmic function
A function of the form $f\left(x\right)={}^{g}log\left(x\right)$ is called a logarithmic function. In this function, $g>0$ and $g\ne 1$ is a fixed given base.
The graphs of the functions $y={g}^{x}$ en $y={}^{g}log\left(x\right)$ are mirror images of each other with respect to the line $y=x$. They are inverse functions of each other.
You can therefore derive the characteristics of $y={}^{g}log\left(x\right)$ from those of $y={g}^{x}$:
the domain is $\u27e8\mathrm{0,}\to \u27e9$;
the range is $\mathbb{R}$;
the graph is increasing while $g>1$, and decreasing while $0<g<1$;
the $y$ -axis is the vertical asymptote of the graph.
All functions that can be derived from transformations of $y={}^{g}log\left(x\right)$ are called logarithmic functions.