Precalculus Examples

$\log_{6} (\frac{36 \sqrt[5]{x + 3}}{6 x^{2} - 13 x - 5})$

Step 1

Rewrite $\log_{6} (\frac{36 \sqrt[5]{x + 3}}{6 x^{2} - 13 x - 5})$ as $\log_{6} (36 \sqrt[5]{x + 3}) - \log_{6} (6 x^{2} - 13 x - 5)$ .

$\log_{6} (36 \sqrt[5]{x + 3}) - \log_{6} (6 x^{2} - 13 x - 5)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x + 3}$ as ${(x + 3)}^{\frac{1}{5}}$ .

$\log_{6} (36 {(x + 3)}^{\frac{1}{5}}) - \log_{6} (6 x^{2} - 13 x - 5)$

Step 3

Rewrite $\log_{6} (36 {(x + 3)}^{\frac{1}{5}})$ as $\log_{6} (36) + \log_{6} ({(x + 3)}^{\frac{1}{5}})$ .

$\log_{6} (36) + \log_{6} ({(x + 3)}^{\frac{1}{5}}) - \log_{6} (6 x^{2} - 13 x - 5)$

Step 4

Expand $\log_{6} ({(x + 3)}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\log_{6} (36) + \frac{1}{5} \log_{6} (x + 3) - \log_{6} (6 x^{2} - 13 x - 5)$

Step 5

Logarithm base $6$ of $36$ is $2$ .

$2 + \frac{1}{5} \log_{6} (x + 3) - \log_{6} (6 x^{2} - 13 x - 5)$

Step 6

Combine $\frac{1}{5}$ and $\log_{6} (x + 3)$ .

$2 + \frac{\log_{6} (x + 3)}{5} - \log_{6} (6 x^{2} - 13 x - 5)$

Step 7

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$2 \cdot \frac{5}{5} + \frac{\log_{6} (x + 3)}{5} - \log_{6} (6 x^{2} - 13 x - 5)$

Step 8

Combine $2$ and $\frac{5}{5}$ .

$\frac{2 \cdot 5}{5} + \frac{\log_{6} (x + 3)}{5} - \log_{6} (6 x^{2} - 13 x - 5)$

Step 9

Combine the numerators over the common denominator.

$\frac{2 \cdot 5 + \log_{6} (x + 3)}{5} - \log_{6} (6 x^{2} - 13 x - 5)$

Step 10

Multiply $2$ by $5$ .

$\frac{10 + \log_{6} (x + 3)}{5} - \log_{6} (6 x^{2} - 13 x - 5)$

Step 11

To write $- \log_{6} (6 x^{2} - 13 x - 5)$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{10 + \log_{6} (x + 3)}{5} - \log_{6} (6 x^{2} - 13 x - 5) \cdot \frac{5}{5}$

Step 12

Combine $- \log_{6} (6 x^{2} - 13 x - 5)$ and $\frac{5}{5}$ .

$\frac{10 + \log_{6} (x + 3)}{5} + \frac{- \log_{6} (6 x^{2} - 13 x - 5) \cdot 5}{5}$

Step 13

Combine the numerators over the common denominator.

$\frac{10 + \log_{6} (x + 3) - \log_{6} (6 x^{2} - 13 x - 5) \cdot 5}{5}$

Step 14

Multiply $5$ by $- 1$ .

$\frac{10 + \log_{6} (x + 3) - 5 \log_{6} (6 x^{2} - 13 x - 5)}{5}$