Solving Exponential Equations using Logarithms

To solve an exponential equation:

$1\right)$ Isolate the exponential expression.

$2\right)$ Take the logarithms of both sides.

$3\right)$ Solve for the variable .

Example 1:

Solve for $x$ : ${2}^{x}=12$

$\begin{array}{l}\mathrm{log}{2}^{x}=\mathrm{log}12\\ x\mathrm{log}2=\mathrm{log}12\\ x=\frac{\mathrm{log}12}{\mathrm{log}2}\approx 3.585\end{array}$

Example 2:

Solve for $x$ : $8\left({10}^{x}\right)=12$

$\begin{array}{l}{10}^{x}=\frac{12}{8}=\frac{3}{2}\\ \mathrm{log}{10}^{x}=\mathrm{log}\frac{3}{2}\\ x\mathrm{log}10=\mathrm{log}\frac{3}{2}\\ x=\mathrm{log}\frac{3}{2}\approx 0.176\end{array}$

Example 3:

Solve for $x$ : ${e}^{5x}=30$

$\begin{array}{l}\mathrm{ln}{e}^{5x}=\mathrm{ln}30\\ 5x\mathrm{ln}e=\mathrm{ln}30\\ 5x=\mathrm{ln}30\\ x=\frac{\mathrm{ln}30}{5}\approx 0.680\end{array}$