Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply $5$ by $- 1$ .

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

Step 5

Rewrite $\log_{5} (a^{\frac{1}{2}} b^{2})$ as $\log_{5} (a^{\frac{1}{2}}) + \log_{5} (b^{2})$ .

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

Step 6

Expand $\log_{5} (a^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 7

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

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

Step 8

Combine $\frac{1}{2}$ and $\log_{5} (a)$ .

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