Precalculus Examples

$\log_{7} (\frac{\sqrt{a^{2} b}}{c^{5}})$

Step 1

Rewrite $\log_{7} (\frac{\sqrt{a^{2} b}}{c^{5}})$ as $\log_{7} (\sqrt{a^{2} b}) - \log_{7} (c^{5})$ .

$\log_{7} (\sqrt{a^{2} b}) - \log_{7} (c^{5})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a^{2} b}$ as ${(a^{2} b)}^{\frac{1}{2}}$ .

$\log_{7} ({(a^{2} b)}^{\frac{1}{2}}) - \log_{7} (c^{5})$

Step 3

Expand $\log_{7} ({(a^{2} b)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{7} (a^{2} b) - \log_{7} (c^{5})$

Step 4

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

$\frac{1}{2} \log_{7} (a^{2} b) - (5 \log_{7} (c))$

Step 5

Multiply $5$ by $- 1$ .

$\frac{1}{2} \log_{7} (a^{2} b) - 5 \log_{7} (c)$

Step 6

Rewrite $\log_{7} (a^{2} b)$ as $\log_{7} (a^{2}) + \log_{7} (b)$ .

$\frac{1}{2} (\log_{7} (a^{2}) + \log_{7} (b)) - 5 \log_{7} (c)$

Step 7

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

$\frac{1}{2} (2 \log_{7} (a) + \log_{7} (b)) - 5 \log_{7} (c)$

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

$\frac{1}{2} (2 \log_{7} (a)) + \frac{1}{2} \log_{7} (b) - 5 \log_{7} (c)$

Step 8.2

Cancel the common factor of $2$ .

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Step 8.2.1

Factor $2$ out of $2 \log_{7} (a)$ .

$\frac{1}{2} (2 (\log_{7} (a))) + \frac{1}{2} \log_{7} (b) - 5 \log_{7} (c)$

Step 8.2.2

Cancel the common factor.

$\frac{1}{2} (2 \log_{7} (a)) + \frac{1}{2} \log_{7} (b) - 5 \log_{7} (c)$

Step 8.2.3

Rewrite the expression.

$\log_{7} (a) + \frac{1}{2} \log_{7} (b) - 5 \log_{7} (c)$

Step 8.3

Combine $\frac{1}{2}$ and $\log_{7} (b)$ .

$\log_{7} (a) + \frac{\log_{7} (b)}{2} - 5 \log_{7} (c)$