Precalculus Examples

$\log_{a} (\sqrt[5]{a \cdot a^{3}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{a \cdot a^{3}}$ as ${(a \cdot a^{3})}^{\frac{1}{5}}$ .

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

Step 2

Expand $\log_{a} ({(a \cdot a^{3})}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

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

Step 3

Multiply $a$ by $a^{3}$ by adding the exponents.

Tap for more steps...

Step 3.1

Multiply $a$ by $a^{3}$ .

Tap for more steps...

Step 3.1.1

Raise $a$ to the power of $1$ .

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

Step 3.1.2

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

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

Step 3.2

Add $1$ and $3$ .

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

Step 4

Expand $\log_{a} (a^{4})$ by moving $4$ outside the logarithm.

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

Step 5

Logarithm base $a$ of $a$ is $1$ .

$\frac{1}{5} (4 \cdot 1)$

Step 6

Multiply $4$ by $1$ .

$\frac{1}{5} \cdot 4$

Step 7

Combine $\frac{1}{5}$ and $4$ .

$\frac{4}{5}$