Simplify/Condense 1/2( log base 5 of x+ log base 5 of y)-2 log base 5 of x+1

$\frac{1}{2} (\log_{5} (x) + \log_{5} (y)) - 2 \log_{5} (x + 1)$

Simplify each term.

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

$\frac{1}{2} \cdot \log_{5} (x y) - 2 \log_{5} (x + 1)$

Simplify $\frac{1}{2} \log_{5} (x y)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{5} ({(x y)}^{\frac{1}{2}}) - 2 \log_{5} (x + 1)$

Apply the product rule to $x y$ .

$\log_{5} (x^{\frac{1}{2}} y^{\frac{1}{2}}) - 2 \log_{5} (x + 1)$

Simplify $- 2 \log_{5} (x + 1)$ by moving $2$ inside the logarithm.

$\log_{5} (x^{\frac{1}{2}} y^{\frac{1}{2}}) - \log_{5} ({(x + 1)}^{2})$

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

$\log_{5} (\frac{x^{\frac{1}{2}} y^{\frac{1}{2}}}{{(x + 1)}^{2}})$