Precalculus Examples

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

Step 1

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

$\ln (5 x^{5} \sqrt{x^{3} - 4}) - \ln (x^{2})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3} - 4}$ as ${(x^{3} - 4)}^{\frac{1}{2}}$ .

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

Step 3

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

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

Step 4

Multiply $2$ by $- 1$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Expand $\ln ({(x^{3} - 4)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 9

Combine $\frac{1}{2}$ and $\ln (x^{3} - 4)$ .

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

Step 10

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

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

Step 11

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

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

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

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

Step 14

Subtract $2 \ln (x)$ from $5 \ln (x)$ .

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