Precalculus Examples

$\log (\frac{a^{9}}{b^{6} \sqrt{c}})$

Step 1

Rewrite $\log (\frac{a^{9}}{b^{6} \sqrt{c}})$ as $\log (a^{9}) - \log (b^{6} \sqrt{c})$ .

$\log (a^{9}) - \log (b^{6} \sqrt{c})$

Step 2

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

$\log (a^{9}) - \log (b^{6} c^{\frac{1}{2}})$

Step 3

Expand $\log (a^{9})$ by moving $9$ outside the logarithm.

$9 \log (a) - \log (b^{6} c^{\frac{1}{2}})$

Step 4

Rewrite $\log (b^{6} c^{\frac{1}{2}})$ as $\log (b^{6}) + \log (c^{\frac{1}{2}})$ .

$9 \log (a) - (\log (b^{6}) + \log (c^{\frac{1}{2}}))$

Step 5

Expand $\log (b^{6})$ by moving $6$ outside the logarithm.

$9 \log (a) - (6 \log (b) + \log (c^{\frac{1}{2}}))$

Step 6

Expand $\log (c^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$9 \log (a) - (6 \log (b) + \frac{1}{2} \log (c))$

Step 7

Simplify each term.

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Step 7.1

Combine $\frac{1}{2}$ and $\log (c)$ .

$9 \log (a) - (6 \log (b) + \frac{\log (c)}{2})$

Step 7.2

Apply the distributive property.

$9 \log (a) - (6 \log (b)) - \frac{\log (c)}{2}$

Step 7.3

Multiply $6$ by $- 1$ .

$9 \log (a) - 6 \log (b) - \frac{\log (c)}{2}$