Exponential Equations

Let's take a look at some examples to review how to solve exponential equations.

Some equations can be solved by writing each side using a common base.

Example

Solve $\left(\frac{1}{3}\right)^x=81$.

Solution

$\left(\frac{1}{3}\right)^x=81$

$\left(3^{-1}\right)^x=3^4$

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

$-x=4$

$x=-4$

Example

Solve $2^{x+4}=8^{x-6}$

Solution

$2^{x+4}=8^{x-6}$

$2^{x+4}=\left(2^3\right)^{x-6}$

$2^{x+4}=2^{3x-18}$

$x+4=3x-18$

$-2x=-22$

$x=11$

Here is an example with a fractional exponent.

Example

Solve $x^{\frac{3}{4}}=81$

Solution

$x^{\frac{3}{4}}=81$

$\left(\sqrt[3]{x}\right)^4=81$

Taking the fourth root of both sides and using a $\pm.$

$\sqrt[3]{x}=\pm3$

Cubing both sides.

$x=\pm27$