Methods for solving exponential inequalities

Inequalities $a^{f(x)}>a^{g(x)}$

If $a>1$ , $a^t$ increases in $t$ , so $a^{f}>a^{g}\ \Leftrightarrow\ f>g$ (on the common domain of the inequality). If $0<a<1$ , the exponential decreases, so $a^{f}>a^{g}\ \Leftrightarrow\ f<g$ . If the bases differ, you usually take logarithms or match bases; with $\ln$ the inequality sign is unchanged ( $\ln$ is increasing), but both sides must be positive.

First ensure $a>0$ , $a\neq1$ , and write the domain for $f,g$

Base	From $a^f>a^g$ you get
$a>1$	$f>g$
$0<a<1$	$f<g$

💡Next lesson — the formal definition of the logarithm and the basic logarithmic identity.

✅You transfer the inequality between $a^{f}$ and $f$ , $g$ according to $a>1$ or $0<a<1$ .