Precalculus Examples

$\log_{5} (\sqrt{8})$

Step 1

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

$\log_{5} (8^{\frac{1}{2}})$

Step 2

Expand $\log_{5} (8^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{5} (8)$

Step 3

Rewrite $\log_{5} (8)$ as $\log_{5} (2^{3})$ .

$\frac{1}{2} \log_{5} (2^{3})$

Step 4

Expand $\log_{5} (2^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} (3 \log_{5} (2))$

Step 5

Multiply $\frac{1}{2} (3 \log_{5} (2))$ .

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Step 5.1

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

$\frac{3}{2} \log_{5} (2)$

Step 5.2

Combine $\frac{3}{2}$ and $\log_{5} (2)$ .

$\frac{3 \log_{5} (2)}{2}$