First try a common base. If that is not useful, isolate the exponential expression and take a logarithm of both positive sides. Example 4x+1=8x-1 22x+2=23x-3 2x+2=3x-3 x=5 Example 3 e2x+1=16 e2x=5 2x=ln5 x= ln52 0.805 Evaluate ln5 with the natural-log key. For another base, use log_7 20= ln20ln7 1.5395. A calculator approximation should agree with the original equation. For rational powers of a positive base, a1/n denotes the positive real root. For example, 813/4=27, whereas solving x4=81 gives both real solutions x=+-3.

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

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

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

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 \(x=\pm3\).

Exponential and Logarithmic Functions

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Solving exponential equations

Solving exponential equations