Precalculus Examples

$\ln (\sqrt{8 x})$

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

$\ln ({(8 x)}^{\frac{1}{2}})$

Expand $\ln ({(8 x)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \ln (8 x)$

Rewrite $\ln (8 x)$ as $\ln (8) + \ln (x)$ .

$\frac{1}{2} (\ln (8) + \ln (x))$

Simplify each term.

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Rewrite $\ln (8)$ as $\ln (2^{3})$ .

$\frac{1}{2} (\ln (2^{3}) + \ln (x))$

Expand $\ln (2^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} (3 \ln (2) + \ln (x))$

Apply the distributive property.

$\frac{1}{2} (3 \ln (2)) + \frac{1}{2} \ln (x)$

Multiply $\frac{1}{2} (3 \ln (2))$ .

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Combine $3$ and $\frac{1}{2}$ .

$\frac{3}{2} \ln (2) + \frac{1}{2} \ln (x)$

Combine $\frac{3}{2}$ and $\ln (2)$ .

$\frac{3 \ln (2)}{2} + \frac{1}{2} \ln (x)$

Combine $\frac{1}{2}$ and $\ln (x)$ .

$\frac{3 \ln (2)}{2} + \frac{\ln (x)}{2}$