# Solving Exponential Equations

Exponential equations are equations in which variables occur as exponents.

For example, exponential equations are in the form ${a}^{x}={b}^{y}$ .

To solve exponential equations with same base, use the property of equality of exponential functions .

If $b$ is a positive number other than $1$ , then ${b}^{x}={b}^{y}$ if and only if $x=y$ . In other words, if the bases are the same, then the exponents must be equal.

Example 1:

Solve the equation ${4}^{2x-1}=64$ .

Note that the bases are not the same. But we can rewrite $64$ as a base of $4$ .

We know that, ${4}^{3}=64$ .

Rewrite $64$ as ${4}^{3}$ so each side has the same base.

${{4}}^{{2}{x}{-}{1}}={{4}}^{{3}}$

By the property of equality of exponential functions, if the bases are the same, then the exponents must be equal.

${2}{x}{-}{1}={3}$

Add $1$ to each side.

$\begin{array}{l}2x-1+1=3+1\\ 2x=4\end{array}$

Divide each side by $2$ .

$\begin{array}{l}\frac{2x}{2}=\frac{4}{2}\\ x=2\end{array}$

Note:

If the bases are not same, then use logarithms to solve the exponential equations. See Solving Exponential Equations using Logarithms .