Expand the Logarithmic Expression natural log of 9/(4x^4y)

$\ln (\frac{9}{4 x^{4} y})$

Step 1

Rewrite $\ln (\frac{9}{4 x^{4} y})$ as $\ln (9) - \ln (4 x^{4} y)$ .

$\ln (9) - \ln (4 x^{4} y)$

Step 2

Rewrite $\ln (4 x^{4} y)$ as $\ln (4 x^{4}) + \ln (y)$ .

$\ln (9) - (\ln (4 x^{4}) + \ln (y))$

Step 3

Rewrite $\ln (4 x^{4})$ as $\ln (4) + \ln (x^{4})$ .

$\ln (9) - (\ln (4) + \ln (x^{4}) + \ln (y))$

Step 4

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

$\ln (9) - (\ln (4) + 4 \ln (x) + \ln (y))$

Step 5

Simplify each term.

Rewrite $\ln (9)$ as $\ln (3^{2})$ .

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

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

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

Simplify each term.

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

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

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

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

Apply the distributive property.

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

Simplify.

Multiply $2$ by $- 1$ .

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

Multiply $4$ by $- 1$ .

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