Logarithmic equations

A logarithmic equation is an equation that involves logarithms of expressions containing a variable. To solve a logarithmic equation, we often use properties of logarithms to combine or simplify terms, and then rewrite the equation in exponential form to solve for the variable.

Important properties

• The logarithm is only defined for positive arguments: $\log_b(x)$ is defined only if $x > 0$ and $b > 0$, $b \neq 1$.

• Key properties: $\log_b(MN) = \log_b M + \log_b N$, $\log_b(M/N) = \log_b M - \log_b N$, $\log_b(M^k) = k \log_b M$.

• To solve $\log_b(x) = c$, rewrite as $x = b^c$.

• Always check that solutions make the arguments of all logarithms positive.