Logarithmic functions > Logarithmic equations
12345Logarithmic equations

## Theory

You can find the solution of a logarithmic equation ${}^{g}log\left(x\right)=a$ by applying an exponential function with base $g$ on both sides.
From ${}^{g}log\left(x\right)=a$ then follows $x={g}^{a}$.
Conditions are that $g>0$ and $g\ne 1$ and $a>0$.

A logarithmic inequality ${}^{g}log\left(x\right) can be solved using graphs:

• First you need to solve the corresponding equation ${}^{g}log\left(x\right)=a$.

• Then you look at the graphs of ${y}_{1}={}^{g}log\left(x\right)$ and ${y}_{2}=a$. Make sure you take into account the domain (and the vertical asymptote) of the logarithm.

• You can read off the solution for the equation.

When equations are more complicated and involve several logarithms you will also need to apply the properties for adding and subtracting logarithms. Sometimes you may also need to change bases.