Precalculus Examples

$\log_{8} (\frac{a ({(b - 2)}^{5})}{6 c^{2}})$

Step 1

Rewrite $\log_{8} (\frac{a ({(b - 2)}^{5})}{6 c^{2}})$ as $\log_{8} (a ({(b - 2)}^{5})) - \log_{8} (6 c^{2})$ .

$\log_{8} (a ({(b - 2)}^{5})) - \log_{8} (6 c^{2})$

Step 2

Multiply $a$ by ${(b - 2)}^{5}$ .

$\log_{8} (a {(b - 2)}^{5}) - \log_{8} (6 c^{2})$

Step 3

Rewrite $\log_{8} (a {(b - 2)}^{5})$ as $\log_{8} (a) + \log_{8} ({(b - 2)}^{5})$ .

$\log_{8} (a) + \log_{8} ({(b - 2)}^{5}) - \log_{8} (6 c^{2})$

Step 4

Expand $\log_{8} ({(b - 2)}^{5})$ by moving $5$ outside the logarithm.

$\log_{8} (a) + 5 \log_{8} (b - 2) - \log_{8} (6 c^{2})$

Step 5

Rewrite $\log_{8} (6 c^{2})$ as $\log_{8} (6) + \log_{8} (c^{2})$ .

$\log_{8} (a) + 5 \log_{8} (b - 2) - (\log_{8} (6) + \log_{8} (c^{2}))$

Step 6

Expand $\log_{8} (c^{2})$ by moving $2$ outside the logarithm.

$\log_{8} (a) + 5 \log_{8} (b - 2) - (\log_{8} (6) + 2 \log_{8} (c))$

Step 7

Simplify each term.

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Step 7.1

Simplify each term.

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Step 7.1.1

Rewrite $\log_{8} (6)$ as $\log_{8} (2 (3))$ .

$\log_{8} (a) + 5 \log_{8} (b - 2) - (\log_{8} (2 (3)) + 2 \log_{8} (c))$

Step 7.1.2

Rewrite $\log_{8} (2 (3))$ as $\log_{8} (2) + \log_{8} (3)$ .

$\log_{8} (a) + 5 \log_{8} (b - 2) - (\log_{8} (2) + \log_{8} (3) + 2 \log_{8} (c))$

Step 7.1.3

Logarithm base $8$ of $2$ is $\frac{1}{3}$ .

$\log_{8} (a) + 5 \log_{8} (b - 2) - (\frac{1}{3} + \log_{8} (3) + 2 \log_{8} (c))$

Step 7.2

Apply the distributive property.

$\log_{8} (a) + 5 \log_{8} (b - 2) - \frac{1}{3} - \log_{8} (3) - (2 \log_{8} (c))$

Step 7.3

Multiply $2$ by $- 1$ .

$\log_{8} (a) + 5 \log_{8} (b - 2) - \frac{1}{3} - \log_{8} (3) - 2 \log_{8} (c)$

Step 8

To write $5 \log_{8} (b - 2)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\log_{8} (a) + 5 \log_{8} (b - 2) \cdot \frac{3}{3} - \frac{1}{3} - \log_{8} (3) - 2 \log_{8} (c)$

Step 9

Combine $5 \log_{8} (b - 2)$ and $\frac{3}{3}$ .

$\log_{8} (a) + \frac{5 \log_{8} (b - 2) \cdot 3}{3} - \frac{1}{3} - \log_{8} (3) - 2 \log_{8} (c)$

Step 10

Combine the numerators over the common denominator.

$\log_{8} (a) + \frac{5 \log_{8} (b - 2) \cdot 3 - 1}{3} - \log_{8} (3) - 2 \log_{8} (c)$

Step 11

Multiply $3$ by $5$ .

$\log_{8} (a) + \frac{15 \log_{8} (b - 2) - 1}{3} - \log_{8} (3) - 2 \log_{8} (c)$