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

For a base \(a>0\), \(a\ne1\), when both \(a^{2x}\) and \(a^x\) occur, let \(u=a^x\), so \(u>0\). Solve the polynomial in \(u\), reject non-positive values and then recover \(x\).

For a logarithmic equation, record the positive-argument restrictions first. Combine logarithms with matching bases, solve the resulting equation and test every candidate in the original domain. Squaring or clearing factors can introduce inadmissible roots.

A graphical solution is an intersection of the two sides or a zero of their difference. Choose a window that respects the domain and check all visible branches; a graph supports a numerical answer but does not replace exact algebra when an exact value is requested.