Expand the Logarithmic Expression natural log of (4xy^2)/15

$\ln (\frac{4 x y^{2}}{15})$

Step 1

Rewrite $\ln (\frac{4 x y^{2}}{15})$ as $\ln (4 x y^{2}) - \ln (15)$ .

$\ln (4 x y^{2}) - \ln (15)$

Step 2

Rewrite $\ln (15)$ as $\ln (3 (5))$ .

$\ln (4 x y^{2}) - \ln (3 (5))$

Step 3

Rewrite $\ln (3 (5))$ as $\ln (3) + \ln (5)$ .

$\ln (4 x y^{2}) - (\ln (3) + \ln (5))$

Step 4

Apply the distributive property.

$\ln (4 x y^{2}) - \ln (3) - \ln (5)$

Step 5

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

$\ln (4 x) + \ln (y^{2}) - \ln (3) - \ln (5)$

Step 6

Rewrite $\ln (4 x)$ as $\ln (4) + \ln (x)$ .

$\ln (4) + \ln (x) + \ln (y^{2}) - \ln (3) - \ln (5)$

Step 7

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

$\ln (4) + \ln (x) + 2 \ln (y) - \ln (3) - \ln (5)$

Step 8

Simplify each term.

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

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

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

$2 \ln (2) + \ln (x) + 2 \ln (y) - \ln (3) - \ln (5)$