Precalculus Examples

$\log_{5} (\frac{a^{2}}{\sqrt[3]{b c}})$

Step 1

Rewrite $\log_{5} (\frac{a^{2}}{\sqrt[3]{b c}})$ as $\log_{5} (a^{2}) - \log_{5} (\sqrt[3]{b c})$ .

$\log_{5} (a^{2}) - \log_{5} (\sqrt[3]{b c})$

Step 2

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

$\log_{5} (a^{2}) - \log_{5} ({(b c)}^{\frac{1}{3}})$

Step 3

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

$2 \log_{5} (a) - \log_{5} ({(b c)}^{\frac{1}{3}})$

Step 4

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

$2 \log_{5} (a) - (\frac{1}{3} \log_{5} (b c))$

Step 5

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

$2 \log_{5} (a) - \frac{\log_{5} (b c)}{3}$

Step 6

Rewrite $\log_{5} (b c)$ as $\log_{5} (b) + \log_{5} (c)$ .

$2 \log_{5} (a) - \frac{\log_{5} (b) + \log_{5} (c)}{3}$

Step 7

To write $2 \log_{5} (a)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 \log_{5} (a) \cdot \frac{3}{3} - \frac{\log_{5} (b) + \log_{5} (c)}{3}$

Step 8

Combine $2 \log_{5} (a)$ and $\frac{3}{3}$ .

$\frac{2 \log_{5} (a) \cdot 3}{3} - \frac{\log_{5} (b) + \log_{5} (c)}{3}$

Step 9

Combine the numerators over the common denominator.

$\frac{2 \log_{5} (a) \cdot 3 - (\log_{5} (b) + \log_{5} (c))}{3}$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $3$ by $2$ .

$\frac{6 \log_{5} (a) - (\log_{5} (b) + \log_{5} (c))}{3}$

Step 10.2

Apply the distributive property.

$\frac{6 \log_{5} (a) - \log_{5} (b) - \log_{5} (c)}{3}$