Precalculus Examples

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{18}$ as $18^{\frac{1}{2}}$ .

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

Step 3

Rewrite $\log_{13} (5 \cdot 18^{\frac{1}{2}})$ as $\log_{13} (5) + \log_{13} (18^{\frac{1}{2}})$ .

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

Step 4

Expand $\log_{13} (18^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Apply the distributive property.

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

Step 9

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

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

Step 10

Cancel the common factor of $2$ .

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Step 10.1

Factor $2$ out of $2 \log_{13} (3)$ .

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

Step 10.2

Cancel the common factor.

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

Step 10.3

Rewrite the expression.

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

Step 11

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

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

Step 12

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

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

Step 13

Simplify each term.

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Step 13.1

Apply the distributive property.

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

Step 13.2

Multiply $2$ by $- 1$ .

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