Simplify/Condense log of x-1/2 log of y+4 log of z

$\log (x) - \frac{1}{2} \log (y) + 4 \log (z)$

Simplify each term.

Multiply $- \frac{1}{2} \log (y)$ .

Reorder $\log (y)$ and $\frac{1}{2}$ .

$\log (x) - (\frac{1}{2} \log (y)) + 4 \log (z)$

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

$\log (x) - \log (y^{\frac{1}{2}}) + 4 \log (z)$

Simplify $4 \log (z)$ by moving $4$ inside the logarithm.

$\log (x) - \log (y^{\frac{1}{2}}) + \log (z^{4})$

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

$\log (\frac{x}{y^{\frac{1}{2}}}) + \log (z^{4})$

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

$\log (\frac{x}{y^{\frac{1}{2}}} z^{4})$

Combine $\frac{x}{y^{\frac{1}{2}}}$ and $z^{4}$ .

$\log (\frac{x z^{4}}{y^{\frac{1}{2}}})$