Precalculus Examples

$\log_{17} (\frac{\sqrt[7]{6}}{y^{2} x})$

Step 1

Rewrite $\log_{17} (\frac{\sqrt[7]{6}}{y^{2} x})$ as $\log_{17} (\sqrt[7]{6}) - \log_{17} (y^{2} x)$ .

$\log_{17} (\sqrt[7]{6}) - \log_{17} (y^{2} x)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{6}$ as $6^{\frac{1}{7}}$ .

$\log_{17} (6^{\frac{1}{7}}) - \log_{17} (y^{2} x)$

Step 3

Expand $\log_{17} (6^{\frac{1}{7}})$ by moving $\frac{1}{7}$ outside the logarithm.

$\frac{1}{7} \log_{17} (6) - \log_{17} (y^{2} x)$

Step 4

Rewrite $\log_{17} (6)$ as $\log_{17} (2 (3))$ .

$\frac{1}{7} \log_{17} (2 (3)) - \log_{17} (y^{2} x)$

Step 5

Rewrite $\log_{17} (2 (3))$ as $\log_{17} (2) + \log_{17} (3)$ .

$\frac{1}{7} (\log_{17} (2) + \log_{17} (3)) - \log_{17} (y^{2} x)$

Step 6

Apply the distributive property.

$\frac{1}{7} \log_{17} (2) + \frac{1}{7} \log_{17} (3) - \log_{17} (y^{2} x)$

Step 7

Combine $\frac{1}{7}$ and $\log_{17} (2)$ .

$\frac{\log_{17} (2)}{7} + \frac{1}{7} \log_{17} (3) - \log_{17} (y^{2} x)$

Step 8

Combine $\frac{1}{7}$ and $\log_{17} (3)$ .

$\frac{\log_{17} (2)}{7} + \frac{\log_{17} (3)}{7} - \log_{17} (y^{2} x)$

Step 9

Rewrite $\log_{17} (y^{2} x)$ as $\log_{17} (y^{2}) + \log_{17} (x)$ .

$\frac{\log_{17} (2)}{7} + \frac{\log_{17} (3)}{7} - (\log_{17} (y^{2}) + \log_{17} (x))$

Step 10

Expand $\log_{17} (y^{2})$ by moving $2$ outside the logarithm.

$\frac{\log_{17} (2)}{7} + \frac{\log_{17} (3)}{7} - (2 \log_{17} (y) + \log_{17} (x))$

Step 11

Simplify each term.

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Step 11.1

Apply the distributive property.

$\frac{\log_{17} (2)}{7} + \frac{\log_{17} (3)}{7} - (2 \log_{17} (y)) - \log_{17} (x)$

Step 11.2

Multiply $2$ by $- 1$ .

$\frac{\log_{17} (2)}{7} + \frac{\log_{17} (3)}{7} - 2 \log_{17} (y) - \log_{17} (x)$