Simplify/Condense 1/3( log of x+ log of y)-4 log of z

$\frac{1}{3} (\log (x) + \log (y)) - 4 \log (z)$

Step 1

Simplify each term.

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

$\frac{1}{3} \cdot \log (x y) - 4 \log (z)$

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

$\log ({(x y)}^{\frac{1}{3}}) - 4 \log (z)$

Apply the product rule to $x y$ .

$\log (x^{\frac{1}{3}} y^{\frac{1}{3}}) - 4 \log (z)$

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

$\log (x^{\frac{1}{3}} y^{\frac{1}{3}}) - \log (z^{4})$

Step 2

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

$\log (\frac{x^{\frac{1}{3}} y^{\frac{1}{3}}}{z^{4}})$