Methods for solving exponential equations

Exponential equations

An equation $a^{f(x)}=a^{g(x)}$ with $a>0$ , $a\neq1$ is equivalent to $f(x)=g(x)$ (monotonicity of the exponential in $x$ ). If the bases differ, try to reduce to one base or take a logarithm of an admissible base (watch the domain). The form $a^{f(x)}=b$ with $b>0$ is solved as $f(x)=\log_a b$ . Sums like $A\cdot a^{2x}+B\cdot a^x+C=0$ are usually turned into a quadratic: $t=a^x>0$ .

After substituting $t=a^x$ , discard roots with $t\le0$

Situation	Step
The same $a$ on both sides	reduce to $a^{f}=a^{g}$ , then $f=g$
$4^x$ and $2^x$	write $4^x=(2^2)^x=2^{2x}$
$3^x+3^{x+1}=12$	factor out $3^x$
Mixture of exponential and constants	$\lg$ / $\ln$ of both sides when the sides are positive

💡Next — exponential inequalities, where the solution hinges on $a>1$ or $0<a<1$ .

✅You distinguish equivalent steps for $a^{f}=a^{g}$ and can reduce a problem to an algebraic equation or to $t=a^x$ .