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

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.

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

Divide each side by $2$ .

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

Note:

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