Precalculus Examples

$\log (\frac{8 x^{2} \sqrt[4]{4 - x}}{9 {(x + 4)}^{2}})$

Step 1

Rewrite $\log (\frac{8 x^{2} \sqrt[4]{4 - x}}{9 {(x + 4)}^{2}})$ as $\log (8 x^{2} \sqrt[4]{4 - x}) - \log (9 {(x + 4)}^{2})$ .

$\log (8 x^{2} \sqrt[4]{4 - x}) - \log (9 {(x + 4)}^{2})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{4 - x}$ as ${(4 - x)}^{\frac{1}{4}}$ .

$\log (8 x^{2} {(4 - x)}^{\frac{1}{4}}) - \log (9 {(x + 4)}^{2})$

Step 3

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

$\log (8 x^{2}) + \log ({(4 - x)}^{\frac{1}{4}}) - \log (9 {(x + 4)}^{2})$

Step 4

Rewrite $\log (8 x^{2})$ as $\log (8) + \log (x^{2})$ .

$\log (8) + \log (x^{2}) + \log ({(4 - x)}^{\frac{1}{4}}) - \log (9 {(x + 4)}^{2})$

Step 5

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

$\log (8) + 2 \log (x) + \log ({(4 - x)}^{\frac{1}{4}}) - \log (9 {(x + 4)}^{2})$

Step 6

Expand $\log ({(4 - x)}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\log (8) + 2 \log (x) + \frac{1}{4} \log (4 - x) - \log (9 {(x + 4)}^{2})$

Step 7

Rewrite $\log (9 {(x + 4)}^{2})$ as $\log (9) + \log ({(x + 4)}^{2})$ .

$\log (8) + 2 \log (x) + \frac{1}{4} \log (4 - x) - (\log (9) + \log ({(x + 4)}^{2}))$

Step 8

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

$\log (8) + 2 \log (x) + \frac{1}{4} \log (4 - x) - (\log (9) + 2 \log (x + 4))$

Step 9

Simplify each term.

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Step 9.1

Rewrite $\log (8)$ as $\log (2^{3})$ .

$\log (2^{3}) + 2 \log (x) + \frac{1}{4} \log (4 - x) - (\log (9) + 2 \log (x + 4))$

Step 9.2

Expand $\log (2^{3})$ by moving $3$ outside the logarithm.

$3 \log (2) + 2 \log (x) + \frac{1}{4} \log (4 - x) - (\log (9) + 2 \log (x + 4))$

Step 9.3

Combine $\frac{1}{4}$ and $\log (4 - x)$ .

$3 \log (2) + 2 \log (x) + \frac{\log (4 - x)}{4} - (\log (9) + 2 \log (x + 4))$

Step 9.4

Simplify each term.

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Step 9.4.1

Rewrite $\log (9)$ as $\log (3^{2})$ .

$3 \log (2) + 2 \log (x) + \frac{\log (4 - x)}{4} - (\log (3^{2}) + 2 \log (x + 4))$

Step 9.4.2

Expand $\log (3^{2})$ by moving $2$ outside the logarithm.

$3 \log (2) + 2 \log (x) + \frac{\log (4 - x)}{4} - (2 \log (3) + 2 \log (x + 4))$

Step 9.5

Apply the distributive property.

$3 \log (2) + 2 \log (x) + \frac{\log (4 - x)}{4} - (2 \log (3)) - (2 \log (x + 4))$

Step 9.6

Multiply $2$ by $- 1$ .

$3 \log (2) + 2 \log (x) + \frac{\log (4 - x)}{4} - 2 \log (3) - (2 \log (x + 4))$

Step 9.7

Multiply $2$ by $- 1$ .

$3 \log (2) + 2 \log (x) + \frac{\log (4 - x)}{4} - 2 \log (3) - 2 \log (x + 4)$

Step 10

To write $3 \log (2)$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 \log (x) + 3 \log (2) \cdot \frac{4}{4} + \frac{\log (4 - x)}{4} - 2 \log (3) - 2 \log (x + 4)$

Step 11

Combine $3 \log (2)$ and $\frac{4}{4}$ .

$2 \log (x) + \frac{3 \log (2) \cdot 4}{4} + \frac{\log (4 - x)}{4} - 2 \log (3) - 2 \log (x + 4)$

Step 12

Combine the numerators over the common denominator.

$2 \log (x) + \frac{3 \log (2) \cdot 4 + \log (4 - x)}{4} - 2 \log (3) - 2 \log (x + 4)$

Step 13

Multiply $4$ by $3$ .

$2 \log (x) + \frac{12 \log (2) + \log (4 - x)}{4} - 2 \log (3) - 2 \log (x + 4)$

Step 14

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

$2 \log (x) + \frac{12 \log (2) + \log (4 - x)}{4} - 2 \log (x + 4) \cdot \frac{4}{4} - 2 \log (3)$

Step 15

Combine $- 2 \log (x + 4)$ and $\frac{4}{4}$ .

$2 \log (x) + \frac{12 \log (2) + \log (4 - x)}{4} + \frac{- 2 \log (x + 4) \cdot 4}{4} - 2 \log (3)$

Step 16

Combine the numerators over the common denominator.

$2 \log (x) + \frac{12 \log (2) + \log (4 - x) - 2 \log (x + 4) \cdot 4}{4} - 2 \log (3)$

Step 17

Multiply $4$ by $- 2$ .

$2 \log (x) + \frac{12 \log (2) + \log (4 - x) - 8 \log (x + 4)}{4} - 2 \log (3)$