Precalculus Examples

$\log (5 \sqrt[3]{5})$

Step 1

Rewrite $\log (5 \sqrt[3]{5})$ as $\log (5) + \log (\sqrt[3]{5})$ .

$\log (5) + \log (\sqrt[3]{5})$

Step 2

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

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

Step 3

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

$\log (5) + \frac{1}{3} \log (5)$

Step 4

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

$\log (5) + \frac{\log (5)}{3}$

Step 5

To write $\log (5)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\log (5) \cdot \frac{3}{3} + \frac{\log (5)}{3}$

Step 6

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

$\frac{\log (5) \cdot 3}{3} + \frac{\log (5)}{3}$

Step 7

Combine the numerators over the common denominator.

$\frac{\log (5) \cdot 3 + \log (5)}{3}$

Step 8

Add $\log (5) \cdot 3$ and $\log (5)$ .

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Step 8.1

Reorder $\log (5)$ and $3$ .

$\frac{3 \cdot \log (5) + \log (5)}{3}$

Step 8.2

Add $3 \cdot \log (5)$ and $\log (5)$ .

$\frac{4 \cdot \log (5)}{3}$

Step 9

Multiply $4$ by $\log (5)$ .

$\frac{4 \log (5)}{3}$