Precalculus Examples

$\log_{16} (x) = - \frac{1}{2}$

Step 1

Rewrite $\log_{16} (x) = - \frac{1}{2}$ 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 $b^{y} = x$ .

$16^{- \frac{1}{2}} = x$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $x = 16^{- \frac{1}{2}}$ .

$x = 16^{- \frac{1}{2}}$

Step 2.2

Simplify $16^{- \frac{1}{2}}$ .

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Step 2.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x = \frac{1}{16^{\frac{1}{2}}}$

Step 2.2.2

Simplify the denominator.

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Step 2.2.2.1

Rewrite $16$ as $4^{2}$ .

$x = \frac{1}{{(4^{2})}^{\frac{1}{2}}}$

Step 2.2.2.2

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

$x = \frac{1}{4^{2 (\frac{1}{2})}}$

Step 2.2.2.3

Cancel the common factor of $2$ .

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Step 2.2.2.3.1

Cancel the common factor.

$x = \frac{1}{4^{2 (\frac{1}{2})}}$

Step 2.2.2.3.2

Rewrite the expression.

$x = \frac{1}{4^{1}}$

Step 2.2.2.4

Evaluate the exponent.

$x = \frac{1}{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{4}$

Decimal Form:

$x = 0.25$