Precalculus Examples

$\log_{5} ({(u \sqrt{v})}^{3})$

Step 1

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

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

Step 2

Expand $\log_{5} ({(u v^{\frac{1}{2}})}^{3})$ by moving $3$ outside the logarithm.

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

Step 3

Rewrite $\log_{5} (u v^{\frac{1}{2}})$ as $\log_{5} (u) + \log_{5} (v^{\frac{1}{2}})$ .

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

Step 4

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

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

Step 5

Combine $\frac{1}{2}$ and $\log_{5} (v)$ .

$3 (\log_{5} (u) + \frac{\log_{5} (v)}{2})$

Step 6

Apply the distributive property.

$3 \log_{5} (u) + 3 \frac{\log_{5} (v)}{2}$

Step 7

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

$3 \log_{5} (u) + \frac{3 \log_{5} (v)}{2}$