Precalculus Examples

$\log_{7} (\frac{\sqrt[5]{17}}{y^{2} x})$

Step 1

Rewrite $\log_{7} (\frac{\sqrt[5]{17}}{y^{2} x})$ as $\log_{7} (\sqrt[5]{17}) - \log_{7} (y^{2} x)$ .

$\log_{7} (\sqrt[5]{17}) - \log_{7} (y^{2} x)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{17}$ as $17^{\frac{1}{5}}$ .

$\log_{7} (17^{\frac{1}{5}}) - \log_{7} (y^{2} x)$

Step 3

Expand $\log_{7} (17^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\frac{1}{5} \log_{7} (17) - \log_{7} (y^{2} x)$

Step 4

Combine $\frac{1}{5}$ and $\log_{7} (17)$ .

$\frac{\log_{7} (17)}{5} - \log_{7} (y^{2} x)$

Step 5

Rewrite $\log_{7} (y^{2} x)$ as $\log_{7} (y^{2}) + \log_{7} (x)$ .

$\frac{\log_{7} (17)}{5} - (\log_{7} (y^{2}) + \log_{7} (x))$

Step 6

Expand $\log_{7} (y^{2})$ by moving $2$ outside the logarithm.

$\frac{\log_{7} (17)}{5} - (2 \log_{7} (y) + \log_{7} (x))$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$\frac{\log_{7} (17)}{5} - (2 \log_{7} (y)) - \log_{7} (x)$

Step 7.2

Multiply $2$ by $- 1$ .

$\frac{\log_{7} (17)}{5} - 2 \log_{7} (y) - \log_{7} (x)$