Precalculus Examples

$\ln (\frac{100 x^{5} \sqrt[3]{5 - x}}{3 {(x + 4)}^{2}})$

Step 1

Rewrite $\ln (\frac{100 x^{5} \sqrt[3]{5 - x}}{3 {(x + 4)}^{2}})$ as $\ln (100 x^{5} \sqrt[3]{5 - x}) - \ln (3 {(x + 4)}^{2})$ .

$\ln (100 x^{5} \sqrt[3]{5 - x}) - \ln (3 {(x + 4)}^{2})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 - x}$ as ${(5 - x)}^{\frac{1}{3}}$ .

$\ln (100 x^{5} {(5 - x)}^{\frac{1}{3}}) - \ln (3 {(x + 4)}^{2})$

Step 3

Rewrite $\ln (100 x^{5} {(5 - x)}^{\frac{1}{3}})$ as $\ln (100 x^{5}) + \ln ({(5 - x)}^{\frac{1}{3}})$ .

$\ln (100 x^{5}) + \ln ({(5 - x)}^{\frac{1}{3}}) - \ln (3 {(x + 4)}^{2})$

Step 4

Rewrite $\ln (100 x^{5})$ as $\ln (100) + \ln (x^{5})$ .

$\ln (100) + \ln (x^{5}) + \ln ({(5 - x)}^{\frac{1}{3}}) - \ln (3 {(x + 4)}^{2})$

Step 5

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

$\ln (100) + 5 \ln (x) + \ln ({(5 - x)}^{\frac{1}{3}}) - \ln (3 {(x + 4)}^{2})$

Step 6

Expand $\ln ({(5 - x)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\ln (100) + 5 \ln (x) + \frac{1}{3} \ln (5 - x) - \ln (3 {(x + 4)}^{2})$

Step 7

Rewrite $\ln (3 {(x + 4)}^{2})$ as $\ln (3) + \ln ({(x + 4)}^{2})$ .

$\ln (100) + 5 \ln (x) + \frac{1}{3} \ln (5 - x) - (\ln (3) + \ln ({(x + 4)}^{2}))$

Step 8

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

$\ln (100) + 5 \ln (x) + \frac{1}{3} \ln (5 - x) - (\ln (3) + 2 \ln (x + 4))$

Step 9

Simplify each term.

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Step 9.1

Rewrite $\ln (100)$ as $\ln (2^{2} \cdot 5^{2})$ .

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

Step 9.2

Rewrite $\ln (2^{2} \cdot 5^{2})$ as $\ln (2^{2}) + \ln (5^{2})$ .

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

Step 9.3

Simplify each term.

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Step 9.3.1

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

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

Step 9.3.2

Expand $\ln (5^{2})$ by moving $2$ outside the logarithm.

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

Step 9.4

Combine $\frac{1}{3}$ and $\ln (5 - x)$ .

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

Step 9.5

Apply the distributive property.

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

Step 9.6

Multiply $2$ by $- 1$ .

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

Step 10

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

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

Step 11

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

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

Multiply $3$ by $2$ .

$2 \ln (5) + 5 \ln (x) + \frac{6 \ln (2) + \ln (5 - x)}{3} - \ln (3) - 2 \ln (x + 4)$

Step 14

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

$2 \ln (5) + 5 \ln (x) + \frac{6 \ln (2) + \ln (5 - x)}{3} - 2 \ln (x + 4) \cdot \frac{3}{3} - \ln (3)$

Step 15

Combine $- 2 \ln (x + 4)$ and $\frac{3}{3}$ .

$2 \ln (5) + 5 \ln (x) + \frac{6 \ln (2) + \ln (5 - x)}{3} + \frac{- 2 \ln (x + 4) \cdot 3}{3} - \ln (3)$

Step 16

Combine the numerators over the common denominator.

$2 \ln (5) + 5 \ln (x) + \frac{6 \ln (2) + \ln (5 - x) - 2 \ln (x + 4) \cdot 3}{3} - \ln (3)$

Step 17

Multiply $3$ by $- 2$ .

$2 \ln (5) + 5 \ln (x) + \frac{6 \ln (2) + \ln (5 - x) - 6 \ln (x + 4)}{3} - \ln (3)$