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

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

Step 1

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.

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

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

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

Step 2

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

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

Step 3

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

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

Step 4

Combine $\frac{x^{3}}{y^{4}}$ and $z^{\frac{1}{2}}$ .

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