Precalculus Examples

$\log (\frac{2 x^{5} \sqrt[3]{1 - x}}{3 {(x + 1)}^{2}})$

Step 1

Rewrite $\log (\frac{2 x^{5} \sqrt[3]{1 - x}}{3 {(x + 1)}^{2}})$ as $\log (2 x^{5} \sqrt[3]{1 - x}) - \log (3 {(x + 1)}^{2})$ .

$\log (2 x^{5} \sqrt[3]{1 - x}) - \log (3 {(x + 1)}^{2})$

Step 2

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

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

Step 3

Rewrite $\log (2 x^{5} {(1 - x)}^{\frac{1}{3}})$ as $\log (2 x^{5}) + \log ({(1 - x)}^{\frac{1}{3}})$ .

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

Step 4

Rewrite $\log (2 x^{5})$ as $\log (2) + \log (x^{5})$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify each term.

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Step 9.1

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

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Multiply $2$ by $- 1$ .

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

Step 10

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

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

Step 11

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

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

Move $3$ to the left of $\log (2)$ .

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

Step 14

To write $- 2 \log (x + 1)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 15

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

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

Step 16

Combine the numerators over the common denominator.

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

Step 17

Multiply $3$ by $- 2$ .

$5 \log (x) + \frac{3 \log (2) + \log (1 - x) - 6 \log (x + 1)}{3} - \log (3)$