Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply $7$ by $- 1$ .

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

Step 5

Rewrite $\log_{g} (a^{\frac{1}{2}} b^{7})$ as $\log_{g} (a^{\frac{1}{2}}) + \log_{g} (b^{7})$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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