Solving exponential equations

First try a common base. If that is not useful, isolate the exponential expression and take a logarithm of both positive sides.

Example

\(\begin{aligned} 4^{x+1}&=8^{x-1}\\2^{2x+2}&=2^{3x-3}\\2x+2&=3x-3\\x&=5\end{aligned}\)

Example

\(\begin{aligned}3\mathrm e^{2x}+1&=16\\\mathrm e^{2x}&=5\\2x&=\ln5\\x&=\dfrac{\ln5}{2}\approx0.805\end{aligned}\)

Evaluate \(\ln5\) with the natural-log key. For another base, use \(\log_7 20=\dfrac{\ln20}{\ln7}\approx1.5395\). A calculator approximation should agree with the original equation.

For rational powers of a positive base, \(a^{1/n}\) denotes the positive real root. For example, \(81^{3/4}=27\), whereas solving \(x^4=81\) gives both real solutions \(x=\pm3\).