# Solving Logarithmic Equations

Logarithmic equations come in two versions: ones with one logarithm and ones with more than one logarithm. We'll cover both types in the lectures below.

Solving logarithmic equations relies heavily on logarithmic properties and the definition of a logarithm. The definition, in particular, will give us a way to get variables out of a logarithm by converting the equation to to an equivalent exponential equation.

Definition of a Logarithm |
$y = \log_a x \Leftrightarrow a_y = x$ |

Properties of Logarithms |
$\log_bx + \log_by = \log_b(xy)$ $$\log_bx - \log_by = \log_b\left(\frac{x}{y}\right)$$ $$m\log_bx = \log_b x^m$$ |

# Video Lectures

Solving equations with two or more logarithms works very similarly to the single logarithm case. We're just going to start by using the properties of logarithms to combine the multiple logarithms into one then the process is identical to what you do when there's just one logarithm.

To solve the simplest kind of logarithmic equation, one with a single logarithm in it, we're going to rely on the definition of a logarithm, i.e. we're going to start by converting the logarithmic equation to an equivalent exponential one.