Expand the Logarithmic Expression natural log of (5p^3q^4)^2

$\ln ({(5 p^{3} q^{4})}^{2})$

Expand $\ln ({(5 p^{3} q^{4})}^{2})$ by moving $2$ outside the logarithm.

$2 \ln (5 p^{3} q^{4})$

Rewrite $\ln (5 p^{3} q^{4})$ as $\ln (5 p^{3}) + \ln (q^{4})$ .

$2 (\ln (5 p^{3}) + \ln (q^{4}))$

Rewrite $\ln (5 p^{3})$ as $\ln (5) + \ln (p^{3})$ .

$2 (\ln (5) + \ln (p^{3}) + \ln (q^{4}))$

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

$2 (\ln (5) + 3 \ln (p) + \ln (q^{4}))$

Expand $\ln (q^{4})$ by moving $4$ outside the logarithm.

$2 (\ln (5) + 3 \ln (p) + 4 \ln (q))$

Apply the distributive property.

$2 \ln (5) + 2 (3 \ln (p)) + 2 (4 \ln (q))$

Simplify.

Multiply $3$ by $2$ .

$2 \ln (5) + 6 \ln (p) + 2 (4 \ln (q))$

Multiply $4$ by $2$ .

$2 \ln (5) + 6 \ln (p) + 8 \ln (q)$