Calculus Examples

$\ln (\frac{7 {(x^{2} - 3)}^{4} e^{5 x}}{\sqrt[3]{x^{3} - 4}})$

Step 1

Rewrite $\ln (\frac{7 {(x^{2} - 3)}^{4} e^{5 x}}{\sqrt[3]{x^{3} - 4}})$ as $\ln (7 {(x^{2} - 3)}^{4} e^{5 x}) - \ln (\sqrt[3]{x^{3} - 4})$ .

$\ln (7 {(x^{2} - 3)}^{4} e^{5 x}) - \ln (\sqrt[3]{x^{3} - 4})$

Step 2

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

$\ln (7 {(x^{2} - 3)}^{4} e^{5 x}) - \ln ({(x^{3} - 4)}^{\frac{1}{3}})$

Step 3

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

$\ln (7 {(x^{2} - 3)}^{4} e^{5 x}) - (\frac{1}{3} \ln (x^{3} - 4))$

Step 4

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

$\ln (7 {(x^{2} - 3)}^{4} e^{5 x}) - \frac{\ln (x^{3} - 4)}{3}$

Step 5

Rewrite $\ln (7 {(x^{2} - 3)}^{4} e^{5 x})$ as $\ln (7 {(x^{2} - 3)}^{4}) + \ln (e^{5 x})$ .

$\ln (7 {(x^{2} - 3)}^{4}) + \ln (e^{5 x}) - \frac{\ln (x^{3} - 4)}{3}$

Step 6

Rewrite $\ln (7 {(x^{2} - 3)}^{4})$ as $\ln (7) + \ln ({(x^{2} - 3)}^{4})$ .

$\ln (7) + \ln ({(x^{2} - 3)}^{4}) + \ln (e^{5 x}) - \frac{\ln (x^{3} - 4)}{3}$

Step 7

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

$\ln (7) + 4 \ln (x^{2} - 3) + \ln (e^{5 x}) - \frac{\ln (x^{3} - 4)}{3}$

Step 8

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

$\ln (7) + 4 \ln (x^{2} - 3) + 5 x \ln (e) - \frac{\ln (x^{3} - 4)}{3}$

Step 9

The natural logarithm of $e$ is $1$ .

$\ln (7) + 4 \ln (x^{2} - 3) + 5 x \cdot 1 - \frac{\ln (x^{3} - 4)}{3}$

Step 10

Multiply $5$ by $1$ .

$\ln (7) + 4 \ln (x^{2} - 3) + 5 x - \frac{\ln (x^{3} - 4)}{3}$

Step 11

To write $\ln (7)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 \ln (x^{2} - 3) + 5 x + \ln (7) \cdot \frac{3}{3} - \frac{\ln (x^{3} - 4)}{3}$

Step 12

Combine $\ln (7)$ and $\frac{3}{3}$ .

$4 \ln (x^{2} - 3) + 5 x + \frac{\ln (7) \cdot 3}{3} - \frac{\ln (x^{3} - 4)}{3}$

Step 13

Combine the numerators over the common denominator.

$4 \ln (x^{2} - 3) + 5 x + \frac{\ln (7) \cdot 3 - \ln (x^{3} - 4)}{3}$

Step 14

Move $3$ to the left of $\ln (7)$ .

$4 \ln (x^{2} - 3) + 5 x + \frac{3 \ln (7) - \ln (x^{3} - 4)}{3}$

Step 15

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

$5 x + 4 \ln (x^{2} - 3) \cdot \frac{3}{3} + \frac{3 \ln (7) - \ln (x^{3} - 4)}{3}$

Step 16

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

$5 x + \frac{4 \ln (x^{2} - 3) \cdot 3}{3} + \frac{3 \ln (7) - \ln (x^{3} - 4)}{3}$

Step 17

Combine the numerators over the common denominator.

$5 x + \frac{4 \ln (x^{2} - 3) \cdot 3 + 3 \ln (7) - \ln (x^{3} - 4)}{3}$

Step 18

Multiply $3$ by $4$ .

$5 x + \frac{12 \ln (x^{2} - 3) + 3 \ln (7) - \ln (x^{3} - 4)}{3}$