Precalculus Examples

$\log_{13} (\frac{\sqrt[5]{18}}{s^{2} r})$

Step 1

Rewrite $\log_{13} (\frac{\sqrt[5]{18}}{s^{2} r})$ as $\log_{13} (\sqrt[5]{18}) - \log_{13} (s^{2} r)$ .

$\log_{13} (\sqrt[5]{18}) - \log_{13} (s^{2} r)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{18}$ as $18^{\frac{1}{5}}$ .

$\log_{13} (18^{\frac{1}{5}}) - \log_{13} (s^{2} r)$

Step 3

Expand $\log_{13} (18^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\frac{1}{5} \log_{13} (18) - \log_{13} (s^{2} r)$

Step 4

Rewrite $\log_{13} (18)$ as $\log_{13} (2 \cdot 3^{2})$ .

$\frac{1}{5} \log_{13} (2 \cdot 3^{2}) - \log_{13} (s^{2} r)$

Step 5

Rewrite $\log_{13} (2 \cdot 3^{2})$ as $\log_{13} (2) + \log_{13} (3^{2})$ .

$\frac{1}{5} (\log_{13} (2) + \log_{13} (3^{2})) - \log_{13} (s^{2} r)$

Step 6

Expand $\log_{13} (3^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{5} (\log_{13} (2) + 2 \log_{13} (3)) - \log_{13} (s^{2} r)$

Step 7

Apply the distributive property.

$\frac{1}{5} \log_{13} (2) + \frac{1}{5} (2 \log_{13} (3)) - \log_{13} (s^{2} r)$

Step 8

Combine $\frac{1}{5}$ and $\log_{13} (2)$ .

$\frac{\log_{13} (2)}{5} + \frac{1}{5} (2 \log_{13} (3)) - \log_{13} (s^{2} r)$

Step 9

Multiply $\frac{1}{5} (2 \log_{13} (3))$ .

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Step 9.1

Combine $2$ and $\frac{1}{5}$ .

$\frac{\log_{13} (2)}{5} + \frac{2}{5} \log_{13} (3) - \log_{13} (s^{2} r)$

Step 9.2

Combine $\frac{2}{5}$ and $\log_{13} (3)$ .

$\frac{\log_{13} (2)}{5} + \frac{2 \log_{13} (3)}{5} - \log_{13} (s^{2} r)$

Step 10

Rewrite $\log_{13} (s^{2} r)$ as $\log_{13} (s^{2}) + \log_{13} (r)$ .

$\frac{\log_{13} (2)}{5} + \frac{2 \log_{13} (3)}{5} - (\log_{13} (s^{2}) + \log_{13} (r))$

Step 11

Expand $\log_{13} (s^{2})$ by moving $2$ outside the logarithm.

$\frac{\log_{13} (2)}{5} + \frac{2 \log_{13} (3)}{5} - (2 \log_{13} (s) + \log_{13} (r))$

Step 12

Simplify each term.

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Step 12.1

Apply the distributive property.

$\frac{\log_{13} (2)}{5} + \frac{2 \log_{13} (3)}{5} - (2 \log_{13} (s)) - \log_{13} (r)$

Step 12.2

Multiply $2$ by $- 1$ .

$\frac{\log_{13} (2)}{5} + \frac{2 \log_{13} (3)}{5} - 2 \log_{13} (s) - \log_{13} (r)$