Simplify/Condense 3 log base 2 of x-4 log base 2 of x+3+ log base 2 of y

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

Step 1

Simplify each term.

Simplify $3 \log_{2} (x)$ by moving $3$ inside the logarithm.

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

Simplify $- 4 \log_{2} (x + 3)$ by moving $4$ inside the logarithm.

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

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{2} (\frac{x^{3}}{{(x + 3)}^{4}}) + \log_{2} (y)$

Step 3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{2} (\frac{x^{3}}{{(x + 3)}^{4}} y)$

Step 4

Combine $\frac{x^{3}}{{(x + 3)}^{4}}$ and $y$ .

$\log_{2} (\frac{x^{3} y}{{(x + 3)}^{4}})$