Precalculus Examples

$\log (\frac{100 x^{3} \sqrt[3]{2 - x}}{7 (x + 6)})$

Step 1

Rewrite $\log (\frac{100 x^{3} \sqrt[3]{2 - x}}{7 (x + 6)})$ as $\log (100 x^{3} \sqrt[3]{2 - x}) - \log (7 (x + 6))$ .

$\log (100 x^{3} \sqrt[3]{2 - x}) - \log (7 (x + 6))$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 - x}$ as ${(2 - x)}^{\frac{1}{3}}$ .

$\log (100 x^{3} {(2 - x)}^{\frac{1}{3}}) - \log (7 (x + 6))$

Step 3

Rewrite $\log (100 x^{3} {(2 - x)}^{\frac{1}{3}})$ as $\log (100 x^{3}) + \log ({(2 - x)}^{\frac{1}{3}})$ .

$\log (100 x^{3}) + \log ({(2 - x)}^{\frac{1}{3}}) - \log (7 (x + 6))$

Step 4

Rewrite $\log (100 x^{3})$ as $\log (100) + \log (x^{3})$ .

$\log (100) + \log (x^{3}) + \log ({(2 - x)}^{\frac{1}{3}}) - \log (7 (x + 6))$

Step 5

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

$\log (100) + 3 \log (x) + \log ({(2 - x)}^{\frac{1}{3}}) - \log (7 (x + 6))$

Step 6

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

$\log (100) + 3 \log (x) + \frac{1}{3} \log (2 - x) - \log (7 (x + 6))$

Step 7

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

$2 + 3 \log (x) + \frac{1}{3} \log (2 - x) - \log (7 (x + 6))$

Step 8

Rewrite $\log (7 (x + 6))$ as $\log (7) + \log (x + 6)$ .

$2 + 3 \log (x) + \frac{1}{3} \log (2 - x) - (\log (7) + \log (x + 6))$

Step 9

Simplify each term.

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Step 9.1

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

$2 + 3 \log (x) + \frac{\log (2 - x)}{3} - (\log (7) + \log (x + 6))$

Step 9.2

Apply the distributive property.

$2 + 3 \log (x) + \frac{\log (2 - x)}{3} - \log (7) - \log (x + 6)$

Step 10

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

$3 \log (x) + 2 \cdot \frac{3}{3} + \frac{\log (2 - x)}{3} - \log (7) - \log (x + 6)$

Step 11

Combine $2$ and $\frac{3}{3}$ .

$3 \log (x) + \frac{2 \cdot 3}{3} + \frac{\log (2 - x)}{3} - \log (7) - \log (x + 6)$

Step 12

Combine the numerators over the common denominator.

$3 \log (x) + \frac{2 \cdot 3 + \log (2 - x)}{3} - \log (7) - \log (x + 6)$

Step 13

Multiply $2$ by $3$ .

$3 \log (x) + \frac{6 + \log (2 - x)}{3} - \log (7) - \log (x + 6)$

Step 14

To write $- \log (x + 6)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 \log (x) + \frac{6 + \log (2 - x)}{3} - \log (x + 6) \cdot \frac{3}{3} - \log (7)$

Step 15

Combine $- \log (x + 6)$ and $\frac{3}{3}$ .

$3 \log (x) + \frac{6 + \log (2 - x)}{3} + \frac{- \log (x + 6) \cdot 3}{3} - \log (7)$

Step 16

Combine the numerators over the common denominator.

$3 \log (x) + \frac{6 + \log (2 - x) - \log (x + 6) \cdot 3}{3} - \log (7)$

Step 17

Multiply $3$ by $- 1$ .

$3 \log (x) + \frac{6 + \log (2 - x) - 3 \log (x + 6)}{3} - \log (7)$