Precalculus Examples

$\log_{3} (\frac{5 \sqrt[8]{x - 5}}{27 x^{9}})$

Step 1

Rewrite $\log_{3} (\frac{5 \sqrt[8]{x - 5}}{27 x^{9}})$ as $\log_{3} (5 \sqrt[8]{x - 5}) - \log_{3} (27 x^{9})$ .

$\log_{3} (5 \sqrt[8]{x - 5}) - \log_{3} (27 x^{9})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[8]{x - 5}$ as ${(x - 5)}^{\frac{1}{8}}$ .

$\log_{3} (5 {(x - 5)}^{\frac{1}{8}}) - \log_{3} (27 x^{9})$

Step 3

Rewrite $\log_{3} (5 {(x - 5)}^{\frac{1}{8}})$ as $\log_{3} (5) + \log_{3} ({(x - 5)}^{\frac{1}{8}})$ .

$\log_{3} (5) + \log_{3} ({(x - 5)}^{\frac{1}{8}}) - \log_{3} (27 x^{9})$

Step 4

Expand $\log_{3} ({(x - 5)}^{\frac{1}{8}})$ by moving $\frac{1}{8}$ outside the logarithm.

$\log_{3} (5) + \frac{1}{8} \log_{3} (x - 5) - \log_{3} (27 x^{9})$

Step 5

Rewrite $\log_{3} (27 x^{9})$ as $\log_{3} (27) + \log_{3} (x^{9})$ .

$\log_{3} (5) + \frac{1}{8} \log_{3} (x - 5) - (\log_{3} (27) + \log_{3} (x^{9}))$

Step 6

Expand $\log_{3} (x^{9})$ by moving $9$ outside the logarithm.

$\log_{3} (5) + \frac{1}{8} \log_{3} (x - 5) - (\log_{3} (27) + 9 \log_{3} (x))$

Step 7

Logarithm base $3$ of $27$ is $3$ .

$\log_{3} (5) + \frac{1}{8} \log_{3} (x - 5) - (3 + 9 \log_{3} (x))$

Step 8

Simplify each term.

Tap for more steps...

Step 8.1

Combine $\frac{1}{8}$ and $\log_{3} (x - 5)$ .

$\log_{3} (5) + \frac{\log_{3} (x - 5)}{8} - (3 + 9 \log_{3} (x))$

Step 8.2

Apply the distributive property.

$\log_{3} (5) + \frac{\log_{3} (x - 5)}{8} - 1 \cdot 3 - (9 \log_{3} (x))$

Step 8.3

Multiply $- 1$ by $3$ .

$\log_{3} (5) + \frac{\log_{3} (x - 5)}{8} - 3 - (9 \log_{3} (x))$

Step 8.4

Multiply $9$ by $- 1$ .

$\log_{3} (5) + \frac{\log_{3} (x - 5)}{8} - 3 - 9 \log_{3} (x)$

Step 9

To write $\log_{3} (5)$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\log_{3} (5) \cdot \frac{8}{8} + \frac{\log_{3} (x - 5)}{8} - 3 - 9 \log_{3} (x)$

Step 10

Combine $\log_{3} (5)$ and $\frac{8}{8}$ .

$\frac{\log_{3} (5) \cdot 8}{8} + \frac{\log_{3} (x - 5)}{8} - 3 - 9 \log_{3} (x)$

Step 11

Combine the numerators over the common denominator.

$\frac{\log_{3} (5) \cdot 8 + \log_{3} (x - 5)}{8} - 3 - 9 \log_{3} (x)$

Step 12

Move $8$ to the left of $\log_{3} (5)$ .

$\frac{8 \log_{3} (5) + \log_{3} (x - 5)}{8} - 3 - 9 \log_{3} (x)$