Precalculus Examples

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

Step 1

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

$\ln ({(x + 3)}^{4}) - \ln ((x - 5) \sqrt{x + 4})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 4}$ as ${(x + 4)}^{\frac{1}{2}}$ .

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

Step 3

Expand $\ln ({(x + 3)}^{4})$ by moving $4$ outside the logarithm.

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

Step 4

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

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

Step 5

Expand $\ln ({(x + 4)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 6

Simplify each term.

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Step 6.1

Combine $\frac{1}{2}$ and $\ln (x + 4)$ .

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

Step 6.2

To write $\ln (x - 5)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.3

Combine $\ln (x - 5)$ and $\frac{2}{2}$ .

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

Step 6.4

Combine the numerators over the common denominator.

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

Step 6.5

Move $2$ to the left of $\ln (x - 5)$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Multiply $2$ by $4$ .

$\frac{8 \ln (x + 3) - (2 \ln (x - 5) + \ln (x + 4))}{2}$

Step 10.2

Apply the distributive property.

$\frac{8 \ln (x + 3) - (2 \ln (x - 5)) - \ln (x + 4)}{2}$

Step 10.3

Multiply $2$ by $- 1$ .

$\frac{8 \ln (x + 3) - 2 \ln (x - 5) - \ln (x + 4)}{2}$