Take a look at the applet: Logarithm
The solution of the equation ${g}^{x}=a$ is called the logarithm of $a$ to base $g$.
Notation: $x={}^{g}log\left(a\right)$.
Since this equation only has solutions if $0<g<1$ or $g>1$ and if $a>0$, ${}^{g}log\left(a\right)$ can only exist under these conditions.
For the time being you mostly use the intersect function of your graphing calculator
to determine $x={}^{g}log\left(a\right)$.
The general definition for logarithms is:
from ${g}^{x}=y$ follows $x={}^{g}log\left(y\right)$;
from $x={}^{g}log\left(y\right)$ follows ${g}^{x}=y$;
The expressions $x={}^{g}log\left(y\right)$ en ${g}^{x}=y$ are completely equivalent if $0<g<1$ or $g>1$ and if $y>0$.
The exponential function and the logarithm with the same base are called inverse operations, since they reverse the effect of each other.