Precalculus Examples

$\log_{5} (\frac{\sqrt[4]{x} \sqrt[3]{y}}{z^{2}})$

Step 1

Rewrite $\log_{5} (\frac{\sqrt[4]{x} \sqrt[3]{y}}{z^{2}})$ as $\log_{5} (\sqrt[4]{x} \sqrt[3]{y}) - \log_{5} (z^{2})$ .

$\log_{5} (\sqrt[4]{x} \sqrt[3]{y}) - \log_{5} (z^{2})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$\log_{5} (x^{\frac{1}{4}} \sqrt[3]{y}) - \log_{5} (z^{2})$

Step 3

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

$\log_{5} (x^{\frac{1}{4}} y^{\frac{1}{3}}) - \log_{5} (z^{2})$

Step 4

Expand $\log_{5} (z^{2})$ by moving $2$ outside the logarithm.

$\log_{5} (x^{\frac{1}{4}} y^{\frac{1}{3}}) - (2 \log_{5} (z))$

Step 5

Multiply $2$ by $- 1$ .

$\log_{5} (x^{\frac{1}{4}} y^{\frac{1}{3}}) - 2 \log_{5} (z)$

Step 6

Rewrite $\log_{5} (x^{\frac{1}{4}} y^{\frac{1}{3}})$ as $\log_{5} (x^{\frac{1}{4}}) + \log_{5} (y^{\frac{1}{3}})$ .

$\log_{5} (x^{\frac{1}{4}}) + \log_{5} (y^{\frac{1}{3}}) - 2 \log_{5} (z)$

Step 7

Expand $\log_{5} (x^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \log_{5} (x) + \log_{5} (y^{\frac{1}{3}}) - 2 \log_{5} (z)$

Step 8

Expand $\log_{5} (y^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{4} \log_{5} (x) + \frac{1}{3} \log_{5} (y) - 2 \log_{5} (z)$

Step 9

Simplify each term.

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Step 9.1

Combine $\frac{1}{4}$ and $\log_{5} (x)$ .

$\frac{\log_{5} (x)}{4} + \frac{1}{3} \log_{5} (y) - 2 \log_{5} (z)$

Step 9.2

Combine $\frac{1}{3}$ and $\log_{5} (y)$ .

$\frac{\log_{5} (x)}{4} + \frac{\log_{5} (y)}{3} - 2 \log_{5} (z)$