Precalculus Examples

$\log (\frac{\sqrt{a}}{b c^{2}})$

Step 1

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

$\log (\sqrt{a}) - \log (b c^{2})$

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify each term.

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Step 6.1

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $2$ by $- 1$ .

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