Expand the Logarithmic Expression log of (x^5y^-2)/(2y)

$\log (\frac{x^{5} y^{- 2}}{2 y})$

Step 1

Rewrite $\log (\frac{x^{5} y^{- 2}}{2 y})$ as $\log (x^{5} y^{- 2}) - \log (2 y)$ .

$\log (x^{5} y^{- 2}) - \log (2 y)$

Step 2

Rewrite $\log (x^{5} y^{- 2})$ as $\log (x^{5}) + \log (y^{- 2})$ .

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

Step 3

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

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

Step 4

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

$5 \log (x) - 2 \log (y) - \log (2 y)$

Step 5

Rewrite $\log (2 y)$ as $\log (2) + \log (y)$ .

$5 \log (x) - 2 \log (y) - (\log (2) + \log (y))$

Step 6

Apply the distributive property.

$5 \log (x) - 2 \log (y) - \log (2) - \log (y)$

Step 7

Subtract $\log (y)$ from $- 2 \log (y)$ .

$5 \log (x) - \log (2) - 3 \log (y)$