Precalculus Examples

$\log_{5} (\sqrt[4]{15})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{15}$ as $15^{\frac{1}{4}}$ .

$\log_{5} (15^{\frac{1}{4}})$

Step 2

Expand $\log_{5} (15^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \log_{5} (15)$

Step 3

Rewrite $\log_{5} (15)$ as $\log_{5} (3 (5))$ .

$\frac{1}{4} \log_{5} (3 (5))$

Step 4

Rewrite $\log_{5} (3 (5))$ as $\log_{5} (3) + \log_{5} (5)$ .

$\frac{1}{4} (\log_{5} (3) + \log_{5} (5))$

Step 5

Logarithm base $5$ of $5$ is $1$ .

$\frac{1}{4} (\log_{5} (3) + 1)$

Step 6

Apply the distributive property.

$\frac{1}{4} \log_{5} (3) + \frac{1}{4} \cdot 1$

Step 7

Combine $\frac{1}{4}$ and $\log_{5} (3)$ .

$\frac{\log_{5} (3)}{4} + \frac{1}{4} \cdot 1$

Step 8

Multiply $\frac{1}{4}$ by $1$ .

$\frac{\log_{5} (3)}{4} + \frac{1}{4}$