Precalculus Examples

$\log_{4} (\frac{16}{\sqrt{x + 8}})$

Step 1

Multiply $\frac{16}{\sqrt{x + 8}}$ by $\frac{\sqrt{x + 8}}{\sqrt{x + 8}}$ .

$\log_{4} (\frac{16}{\sqrt{x + 8}} \cdot \frac{\sqrt{x + 8}}{\sqrt{x + 8}})$

Step 2

Combine and simplify the denominator.

Tap for more steps...

Step 2.1

Multiply $\frac{16}{\sqrt{x + 8}}$ by $\frac{\sqrt{x + 8}}{\sqrt{x + 8}}$ .

$\log_{4} (\frac{16 \sqrt{x + 8}}{\sqrt{x + 8} \sqrt{x + 8}})$

Step 2.2

Raise $\sqrt{x + 8}$ to the power of $1$ .

$\log_{4} (\frac{16 \sqrt{x + 8}}{{\sqrt{x + 8}}^{1} \sqrt{x + 8}})$

Step 2.3

Raise $\sqrt{x + 8}$ to the power of $1$ .

$\log_{4} (\frac{16 \sqrt{x + 8}}{{\sqrt{x + 8}}^{1} {\sqrt{x + 8}}^{1}})$

Step 2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{4} (\frac{16 \sqrt{x + 8}}{{\sqrt{x + 8}}^{1 + 1}})$

Step 2.5

Add $1$ and $1$ .

$\log_{4} (\frac{16 \sqrt{x + 8}}{{\sqrt{x + 8}}^{2}})$

Step 2.6

Rewrite ${\sqrt{x + 8}}^{2}$ as $x + 8$ .

Tap for more steps...

Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 8}$ as ${(x + 8)}^{\frac{1}{2}}$ .

$\log_{4} (\frac{16 \sqrt{x + 8}}{{({(x + 8)}^{\frac{1}{2}})}^{2}})$

Step 2.6.2

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

$\log_{4} (\frac{16 \sqrt{x + 8}}{{(x + 8)}^{\frac{1}{2} \cdot 2}})$

Step 2.6.3

Combine $\frac{1}{2}$ and $2$ .

$\log_{4} (\frac{16 \sqrt{x + 8}}{{(x + 8)}^{\frac{2}{2}}})$

Step 2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.6.4.1

Cancel the common factor.

$\log_{4} (\frac{16 \sqrt{x + 8}}{{(x + 8)}^{\frac{2}{2}}})$

Step 2.6.4.2

Rewrite the expression.

$\log_{4} (\frac{16 \sqrt{x + 8}}{{(x + 8)}^{1}})$

Step 2.6.5

Simplify.

$\log_{4} (\frac{16 \sqrt{x + 8}}{x + 8})$

Step 3

Rewrite $\log_{4} (\frac{16 \sqrt{x + 8}}{x + 8})$ using the change of base formula.

Tap for more steps...

Step 3.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 3.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (\frac{16 \sqrt{x + 8}}{x + 8})}{\log (4)}$