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

$2 \ln (x) - 3 \ln (y) - 4 \ln (z)$

Step 1

Simplify each term.

Simplify $2 \ln (x)$ by moving $2$ inside the logarithm.

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

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

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

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine.

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

Step 6

Multiply $x^{2}$ by $1$ .

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