Precalculus Examples

$\log_{7} (\frac{\sqrt[3]{y}}{z x^{5}})$

Step 1

Rewrite $\log_{7} (\frac{\sqrt[3]{y}}{z x^{5}})$ as $\log_{7} (\sqrt[3]{y}) - \log_{7} (z x^{5})$ .

$\log_{7} (\sqrt[3]{y}) - \log_{7} (z x^{5})$

Step 2

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

$\log_{7} (y^{\frac{1}{3}}) - \log_{7} (z x^{5})$

Step 3

Expand $\log_{7} (y^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{7} (y) - \log_{7} (z x^{5})$

Step 4

Rewrite $\log_{7} (z x^{5})$ as $\log_{7} (z) + \log_{7} (x^{5})$ .

$\frac{1}{3} \log_{7} (y) - (\log_{7} (z) + \log_{7} (x^{5}))$

Step 5

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

$\frac{1}{3} \log_{7} (y) - (\log_{7} (z) + 5 \log_{7} (x))$

Step 6

Simplify each term.

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Step 6.1

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

$\frac{\log_{7} (y)}{3} - (\log_{7} (z) + 5 \log_{7} (x))$

Step 6.2

Apply the distributive property.

$\frac{\log_{7} (y)}{3} - \log_{7} (z) - (5 \log_{7} (x))$

Step 6.3

Multiply $5$ by $- 1$ .

$\frac{\log_{7} (y)}{3} - \log_{7} (z) - 5 \log_{7} (x)$