Expand the Logarithmic Expression log of (5a)/(4b^2)

$\log (\frac{5 a}{4 b^{2}})$

Step 1

Rewrite $\log (\frac{5 a}{4 b^{2}})$ as $\log (5 a) - \log (4 b^{2})$ .

$\log (5 a) - \log (4 b^{2})$

Step 2

Rewrite $\log (5 a)$ as $\log (5) + \log (a)$ .

$\log (5) + \log (a) - \log (4 b^{2})$

Step 3

Rewrite $\log (4 b^{2})$ as $\log (4) + \log (b^{2})$ .

$\log (5) + \log (a) - (\log (4) + \log (b^{2}))$

Step 4

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

$\log (5) + \log (a) - (\log (4) + 2 \log (b))$

Step 5

Simplify each term.

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

$\log (5) + \log (a) - (\log (2^{2}) + 2 \log (b))$

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

$\log (5) + \log (a) - (2 \log (2) + 2 \log (b))$

Apply the distributive property.

$\log (5) + \log (a) - (2 \log (2)) - (2 \log (b))$

Multiply $2$ by $- 1$ .

$\log (5) + \log (a) - 2 \log (2) - (2 \log (b))$

Multiply $2$ by $- 1$ .

$\log (5) + \log (a) - 2 \log (2) - 2 \log (b)$