Expand the Logarithmic Expression natural log of ((2x)/(y^3))^4

$\ln ({(\frac{2 x}{y^{3}})}^{4})$

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

$4 \ln (\frac{2 x}{y^{3}})$

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

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

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

$4 (\ln (2 x) - (3 \ln (y)))$

Multiply $3$ by $- 1$ .

$4 (\ln (2 x) - 3 \ln (y))$

Rewrite $\ln (2 x)$ as $\ln (2) + \ln (x)$ .

$4 (\ln (2) + \ln (x) - 3 \ln (y))$

Apply the distributive property.

$4 \ln (2) + 4 \ln (x) + 4 (- 3 \ln (y))$

Multiply $- 3$ by $4$ .

$4 \ln (2) + 4 \ln (x) - 12 \ln (y)$