Precalculus Examples

$\log (\frac{100 \sqrt[4]{5 - x}}{x^{3} {(x + 7)}^{5}})$

Step 1

Rewrite $\log (\frac{100 \sqrt[4]{5 - x}}{x^{3} {(x + 7)}^{5}})$ as $\log (100 \sqrt[4]{5 - x}) - \log (x^{3} {(x + 7)}^{5})$ .

$\log (100 \sqrt[4]{5 - x}) - \log (x^{3} {(x + 7)}^{5})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{5 - x}$ as ${(5 - x)}^{\frac{1}{4}}$ .

$\log (100 {(5 - x)}^{\frac{1}{4}}) - \log (x^{3} {(x + 7)}^{5})$

Step 3

Rewrite $\log (100 {(5 - x)}^{\frac{1}{4}})$ as $\log (100) + \log ({(5 - x)}^{\frac{1}{4}})$ .

$\log (100) + \log ({(5 - x)}^{\frac{1}{4}}) - \log (x^{3} {(x + 7)}^{5})$

Step 4

Expand $\log ({(5 - x)}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\log (100) + \frac{1}{4} \log (5 - x) - \log (x^{3} {(x + 7)}^{5})$

Step 5

Logarithm base $10$ of $100$ is $2$ .

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

Step 6

Rewrite $\log (x^{3} {(x + 7)}^{5})$ as $\log (x^{3}) + \log ({(x + 7)}^{5})$ .

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

Step 7

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

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

Step 8

Expand $\log ({(x + 7)}^{5})$ by moving $5$ outside the logarithm.

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

Step 9

Simplify each term.

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Step 9.1

Combine $\frac{1}{4}$ and $\log (5 - x)$ .

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Multiply $3$ by $- 1$ .

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

Step 9.4

Multiply $5$ by $- 1$ .

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

Step 10

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

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

Step 11

Combine $2$ and $\frac{4}{4}$ .

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

Multiply $2$ by $4$ .

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

Step 14

To write $- 5 \log (x + 7)$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 15

Combine $- 5 \log (x + 7)$ and $\frac{4}{4}$ .

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

Step 16

Combine the numerators over the common denominator.

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

Step 17

Multiply $4$ by $- 5$ .

$\frac{8 + \log (5 - x) - 20 \log (x + 7)}{4} - 3 \log (x)$