Precalculus Examples

$\log_{2} (\frac{16 x^{4} (\sqrt[3]{x - 8})}{7 {(x + 5)}^{4}})$

Step 1

Rewrite $\log_{2} (\frac{16 x^{4} (\sqrt[3]{x - 8})}{7 {(x + 5)}^{4}})$ as $\log_{2} (16 x^{4} (\sqrt[3]{x - 8})) - \log_{2} (7 {(x + 5)}^{4})$ .

$\log_{2} (16 x^{4} (\sqrt[3]{x - 8})) - \log_{2} (7 {(x + 5)}^{4})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 8}$ as ${(x - 8)}^{\frac{1}{3}}$ .

$\log_{2} (16 x^{4} {(x - 8)}^{\frac{1}{3}}) - \log_{2} (7 {(x + 5)}^{4})$

Step 3

Rewrite $\log_{2} (16 x^{4} {(x - 8)}^{\frac{1}{3}})$ as $\log_{2} (16 x^{4}) + \log_{2} ({(x - 8)}^{\frac{1}{3}})$ .

$\log_{2} (16 x^{4}) + \log_{2} ({(x - 8)}^{\frac{1}{3}}) - \log_{2} (7 {(x + 5)}^{4})$

Step 4

Rewrite $\log_{2} (16 x^{4})$ as $\log_{2} (16) + \log_{2} (x^{4})$ .

$\log_{2} (16) + \log_{2} (x^{4}) + \log_{2} ({(x - 8)}^{\frac{1}{3}}) - \log_{2} (7 {(x + 5)}^{4})$

Step 5

Expand $\log_{2} (x^{4})$ by moving $4$ outside the logarithm.

$\log_{2} (16) + 4 \log_{2} (x) + \log_{2} ({(x - 8)}^{\frac{1}{3}}) - \log_{2} (7 {(x + 5)}^{4})$

Step 6

Expand $\log_{2} ({(x - 8)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\log_{2} (16) + 4 \log_{2} (x) + \frac{1}{3} \log_{2} (x - 8) - \log_{2} (7 {(x + 5)}^{4})$

Step 7

Logarithm base $2$ of $16$ is $4$ .

$4 + 4 \log_{2} (x) + \frac{1}{3} \log_{2} (x - 8) - \log_{2} (7 {(x + 5)}^{4})$

Step 8

Rewrite $\log_{2} (7 {(x + 5)}^{4})$ as $\log_{2} (7) + \log_{2} ({(x + 5)}^{4})$ .

$4 + 4 \log_{2} (x) + \frac{1}{3} \log_{2} (x - 8) - (\log_{2} (7) + \log_{2} ({(x + 5)}^{4}))$

Step 9

Expand $\log_{2} ({(x + 5)}^{4})$ by moving $4$ outside the logarithm.

$4 + 4 \log_{2} (x) + \frac{1}{3} \log_{2} (x - 8) - (\log_{2} (7) + 4 \log_{2} (x + 5))$

Step 10

Simplify each term.

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Step 10.1

Combine $\frac{1}{3}$ and $\log_{2} (x - 8)$ .

$4 + 4 \log_{2} (x) + \frac{\log_{2} (x - 8)}{3} - (\log_{2} (7) + 4 \log_{2} (x + 5))$

Step 10.2

Apply the distributive property.

$4 + 4 \log_{2} (x) + \frac{\log_{2} (x - 8)}{3} - \log_{2} (7) - (4 \log_{2} (x + 5))$

Step 10.3

Multiply $4$ by $- 1$ .

$4 + 4 \log_{2} (x) + \frac{\log_{2} (x - 8)}{3} - \log_{2} (7) - 4 \log_{2} (x + 5)$

Step 11

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 \log_{2} (x) + 4 \cdot \frac{3}{3} + \frac{\log_{2} (x - 8)}{3} - \log_{2} (7) - 4 \log_{2} (x + 5)$

Step 12

Combine $4$ and $\frac{3}{3}$ .

$4 \log_{2} (x) + \frac{4 \cdot 3}{3} + \frac{\log_{2} (x - 8)}{3} - \log_{2} (7) - 4 \log_{2} (x + 5)$

Step 13

Combine the numerators over the common denominator.

$4 \log_{2} (x) + \frac{4 \cdot 3 + \log_{2} (x - 8)}{3} - \log_{2} (7) - 4 \log_{2} (x + 5)$

Step 14

Multiply $4$ by $3$ .

$4 \log_{2} (x) + \frac{12 + \log_{2} (x - 8)}{3} - \log_{2} (7) - 4 \log_{2} (x + 5)$

Step 15

To write $- 4 \log_{2} (x + 5)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 \log_{2} (x) + \frac{12 + \log_{2} (x - 8)}{3} - 4 \log_{2} (x + 5) \cdot \frac{3}{3} - \log_{2} (7)$

Step 16

Combine $- 4 \log_{2} (x + 5)$ and $\frac{3}{3}$ .

$4 \log_{2} (x) + \frac{12 + \log_{2} (x - 8)}{3} + \frac{- 4 \log_{2} (x + 5) \cdot 3}{3} - \log_{2} (7)$

Step 17

Combine the numerators over the common denominator.

$4 \log_{2} (x) + \frac{12 + \log_{2} (x - 8) - 4 \log_{2} (x + 5) \cdot 3}{3} - \log_{2} (7)$

Step 18

Multiply $3$ by $- 4$ .

$4 \log_{2} (x) + \frac{12 + \log_{2} (x - 8) - 12 \log_{2} (x + 5)}{3} - \log_{2} (7)$