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

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

Step 1

Simplify each term.

Simplify $- 3 \ln (y)$ by moving $3$ inside the logarithm.

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

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

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

Step 2

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

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

Step 3

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

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

Step 4

Combine $\frac{x}{y^{3}}$ and $z^{\frac{1}{2}}$ .

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