Logarithmic inequalities

A logarithmic inequality is an inequality that involves a logarithm, such as $\log_a(f(x)) > b$ or $\log_a(f(x)) < g(x)$. To solve these, you often rewrite the inequality in exponential form and consider the domain where the logarithm is defined (i.e., its argument is positive).

Important properties

• The logarithm $\log_a(x)$ is only defined for $x > 0$ and $a > 0$, $a \neq 1$.

• If $a > 1$, $\log_a(x)$ is an increasing function: if $x_1 < x_2$, then $\log_a(x_1) < \log_a(x_2)$.

• If $0 < a < 1$, $\log_a(x)$ is a decreasing function: if $x_1 < x_2$, then $\log_a(x_1) > \log_a(x_2)$.

• To solve $\log_a(f(x)) > b$, rewrite as $f(x) > a^b$ (if $a > 1$) and also require $f(x) > 0$.