Expand the Logarithmic Expression natural log of (8e^2x^5)/((x^2+1) cube root of x+2)

$\ln (\frac{8 e^{2} x^{5}}{(x^{2} + 1) \sqrt[3]{x + 2}})$

Rewrite $\ln (\frac{8 e^{2} x^{5}}{(x^{2} + 1) \sqrt[3]{x + 2}})$ as $\ln (8 e^{2} x^{5}) - \ln ((x^{2} + 1) \sqrt[3]{x + 2})$ .

$\ln (8 e^{2} x^{5}) - \ln ((x^{2} + 1) \sqrt[3]{x + 2})$

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

$\ln (8 e^{2} x^{5}) - \ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})$

Rewrite $\ln (8 e^{2} x^{5})$ as $\ln (8 e^{2}) + \ln (x^{5})$ .

$\ln (8 e^{2}) + \ln (x^{5}) - \ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})$

Rewrite $\ln (8 e^{2})$ as $\ln (8) + \ln (e^{2})$ .

$\ln (8) + \ln (e^{2}) + \ln (x^{5}) - \ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})$

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

$\ln (8) + 2 \ln (e) + \ln (x^{5}) - \ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})$

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

$\ln (8) + 2 \ln (e) + 5 \ln (x) - \ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})$

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

$\ln (8) + 2 \cdot 1 + 5 \ln (x) - \ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})$

Multiply $2$ by $1$ .

$\ln (8) + 2 + 5 \ln (x) - \ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})$

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

$\ln (8) + 2 + 5 \ln (x) - (\ln (x^{2} + 1) + \ln ({(x + 2)}^{\frac{1}{3}}))$

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

$\ln (8) + 2 + 5 \ln (x) - (\ln (x^{2} + 1) + \frac{1}{3} \ln (x + 2))$

Simplify each term.

Rewrite $\ln (8)$ as $\ln (2^{3})$ .

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Move $3$ to the left of $\ln (x^{2} + 1)$ .

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