Precalculus Examples

$\log_{5} (\frac{\sqrt[4]{p} \cdot \sqrt[9]{q}}{t^{2}})$

Step 1

Rewrite $\log_{5} (\frac{\sqrt[4]{p} \cdot \sqrt[9]{q}}{t^{2}})$ as $\log_{5} (\sqrt[4]{p} \cdot \sqrt[9]{q}) - \log_{5} (t^{2})$ .

$\log_{5} (\sqrt[4]{p} \cdot \sqrt[9]{q}) - \log_{5} (t^{2})$

Step 2

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

$\log_{5} (p^{\frac{1}{4}} \cdot \sqrt[9]{q}) - \log_{5} (t^{2})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{q}$ as $q^{\frac{1}{9}}$ .

$\log_{5} (p^{\frac{1}{4}} \cdot q^{\frac{1}{9}}) - \log_{5} (t^{2})$

Step 4

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

$\log_{5} (p^{\frac{1}{4}} \cdot q^{\frac{1}{9}}) - (2 \log_{5} (t))$

Step 5

Multiply $2$ by $- 1$ .

$\log_{5} (p^{\frac{1}{4}} \cdot q^{\frac{1}{9}}) - 2 \log_{5} (t)$

Step 6

Rewrite $\log_{5} (p^{\frac{1}{4}} \cdot q^{\frac{1}{9}})$ as $\log_{5} (p^{\frac{1}{4}}) + \log_{5} (q^{\frac{1}{9}})$ .

$\log_{5} (p^{\frac{1}{4}}) + \log_{5} (q^{\frac{1}{9}}) - 2 \log_{5} (t)$

Step 7

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

$\frac{1}{4} \log_{5} (p) + \log_{5} (q^{\frac{1}{9}}) - 2 \log_{5} (t)$

Step 8

Expand $\log_{5} (q^{\frac{1}{9}})$ by moving $\frac{1}{9}$ outside the logarithm.

$\frac{1}{4} \log_{5} (p) + \frac{1}{9} \log_{5} (q) - 2 \log_{5} (t)$

Step 9

Simplify each term.

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Step 9.1

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

$\frac{\log_{5} (p)}{4} + \frac{1}{9} \log_{5} (q) - 2 \log_{5} (t)$

Step 9.2

Combine $\frac{1}{9}$ and $\log_{5} (q)$ .

$\frac{\log_{5} (p)}{4} + \frac{\log_{5} (q)}{9} - 2 \log_{5} (t)$