Precalculus Examples

$\log_{f} (\frac{\sqrt{a} b^{2}}{c^{4}})$

Step 1

Rewrite $\log_{f} (\frac{\sqrt{a} b^{2}}{c^{4}})$ as $\log_{f} (\sqrt{a} b^{2}) - \log_{f} (c^{4})$ .

$\log_{f} (\sqrt{a} b^{2}) - \log_{f} (c^{4})$

Step 2

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

$\log_{f} (a^{\frac{1}{2}} b^{2}) - \log_{f} (c^{4})$

Step 3

Expand $\log_{f} (c^{4})$ by moving $4$ outside the logarithm.

$\log_{f} (a^{\frac{1}{2}} b^{2}) - (4 \log_{f} (c))$

Step 4

Multiply $4$ by $- 1$ .

$\log_{f} (a^{\frac{1}{2}} b^{2}) - 4 \log_{f} (c)$

Step 5

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

$\log_{f} (a^{\frac{1}{2}}) + \log_{f} (b^{2}) - 4 \log_{f} (c)$

Step 6

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

$\frac{1}{2} \log_{f} (a) + \log_{f} (b^{2}) - 4 \log_{f} (c)$

Step 7

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

$\frac{1}{2} \log_{f} (a) + 2 \log_{f} (b) - 4 \log_{f} (c)$

Step 8

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

$\frac{\log_{f} (a)}{2} + 2 \log_{f} (b) - 4 \log_{f} (c)$