Precalculus Examples

$\log_{32} (x) = \frac{3}{5}$

Step 1

Rewrite $\log_{32} (x) = \frac{3}{5}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$32^{\frac{3}{5}} = x$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $x = 32^{\frac{3}{5}}$ .

$x = 32^{\frac{3}{5}}$

Step 2.2

Simplify $32^{\frac{3}{5}}$ .

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Step 2.2.1

Simplify the expression.

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Step 2.2.1.1

Rewrite $32$ as $2^{5}$ .

$x = {(2^{5})}^{\frac{3}{5}}$

Step 2.2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = 2^{5 (\frac{3}{5})}$

Step 2.2.2

Cancel the common factor of $5$ .

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Step 2.2.2.1

Cancel the common factor.

$x = 2^{5 (\frac{3}{5})}$

Step 2.2.2.2

Rewrite the expression.

$x = 2^{3}$

Step 2.2.3

Raise $2$ to the power of $3$ .

$x = 8$