Write as a Single Logarithm 1+3 natural log of x-4 natural log of y-1/2 natural log of z

$1 + 3 \ln (x) - 4 \ln (y) - \frac{1}{2} \ln (z)$

Simplify each term.

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

$1 + \ln (x^{3}) - 4 \ln (y) - \frac{1}{2} \cdot \ln (z)$

Simplify $- 4 \ln (y)$ by moving $4$ inside the logarithm.

$1 + \ln (x^{3}) - \ln (y^{4}) - \frac{1}{2} \cdot \ln (z)$

Multiply $- \frac{1}{2} \ln (z)$ .

Reorder $\ln (z)$ and $\frac{1}{2}$ .

$1 + \ln (x^{3}) - \ln (y^{4}) - (\frac{1}{2} \ln (z))$

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

$1 + \ln (x^{3}) - \ln (y^{4}) - \ln (z^{\frac{1}{2}})$

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

$1 + \ln (\frac{x^{3}}{y^{4}}) - \ln (z^{\frac{1}{2}})$

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

$1 + \ln (\frac{\frac{x^{3}}{y^{4}}}{z^{\frac{1}{2}}})$

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$1 + \ln (\frac{x^{3}}{y^{4}} \cdot \frac{1}{z^{\frac{1}{2}}})$

Combine.

$1 + \ln (\frac{x^{3} \cdot 1}{y^{4} z^{\frac{1}{2}}})$

Multiply $x^{3}$ by $1$ .

$1 + \ln (\frac{x^{3}}{y^{4} z^{\frac{1}{2}}})$