Expand the Logarithmic Expression natural log of cube root of 5x^5

$\ln (\sqrt[3]{5 x^{5}})$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Apply the distributive property.

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

Step 6

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

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

Step 7

Multiply $\frac{1}{3} (5 \ln (x))$ .

Combine $5$ and $\frac{1}{3}$ .

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

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

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