Precalculus Examples

$\log_{5} (\frac{25}{\sqrt{x + 7}})$

Step 1

Rewrite $\log_{5} (\frac{25}{\sqrt{x + 7}})$ as $\log_{5} (25) - \log_{5} (\sqrt{x + 7})$ .

$\log_{5} (25) - \log_{5} (\sqrt{x + 7})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 7}$ as ${(x + 7)}^{\frac{1}{2}}$ .

$\log_{5} (25) - \log_{5} ({(x + 7)}^{\frac{1}{2}})$

Step 3

Expand $\log_{5} ({(x + 7)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{5} (25) - (\frac{1}{2} \log_{5} (x + 7))$

Step 4

Logarithm base $5$ of $25$ is $2$ .

$2 - (\frac{1}{2} \log_{5} (x + 7))$

Step 5

Combine $\frac{1}{2}$ and $\log_{5} (x + 7)$ .

$2 - \frac{\log_{5} (x + 7)}{2}$

Step 6

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 \cdot \frac{2}{2} - \frac{\log_{5} (x + 7)}{2}$

Step 7

Combine $2$ and $\frac{2}{2}$ .

$\frac{2 \cdot 2}{2} - \frac{\log_{5} (x + 7)}{2}$

Step 8

Combine the numerators over the common denominator.

$\frac{2 \cdot 2 - \log_{5} (x + 7)}{2}$

Step 9

Multiply $2$ by $2$ .

$\frac{4 - \log_{5} (x + 7)}{2}$