Simplify/Condense 1/2* log of x+1/3* log of y-2 log of z

$\frac{1}{2} \cdot \log (x) + \frac{1}{3} \cdot \log (y) - 2 \log (z)$

Simplify each term.

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

$\log (x^{\frac{1}{2}}) + \frac{1}{3} \cdot \log (y) - 2 \log (z)$

Simplify $\frac{1}{3} \log (y)$ by moving $\frac{1}{3}$ inside the logarithm.

$\log (x^{\frac{1}{2}}) + \log (y^{\frac{1}{3}}) - 2 \log (z)$

Simplify $- 2 \log (z)$ by moving $2$ inside the logarithm.

$\log (x^{\frac{1}{2}}) + \log (y^{\frac{1}{3}}) - \log (z^{2})$

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

$\log (x^{\frac{1}{2}} y^{\frac{1}{3}}) - \log (z^{2})$

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

$\log (\frac{x^{\frac{1}{2}} y^{\frac{1}{3}}}{z^{2}})$