Precalculus Examples

$\log (\frac{x^{3} \sqrt[3]{1 - x}}{100 {(x + 2)}^{3}})$

Step 1

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

$\log (x^{3} \sqrt[3]{1 - x}) - \log (100 {(x + 2)}^{3})$

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify each term.

Tap for more steps...

Step 9.1

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

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Multiply $- 1$ by $2$ .

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

Step 9.4

Multiply $3$ by $- 1$ .

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

Step 10

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

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

Step 11

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

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

Multiply $3$ by $3$ .

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

Step 14

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

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

Step 15

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

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

Step 16

Combine the numerators over the common denominator.

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

Step 17

Multiply $3$ by $- 3$ .

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