Evaluate natural log of e^2- natural log of e^11

$\ln (e^{2}) - \ln (e^{11})$

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

$\ln (\frac{e^{2}}{e^{11}})$

Reduce the expression $\frac{e^{2}}{e^{11}}$ by cancelling the common factors.

Multiply by $1$ .

$\ln (\frac{e^{2} \cdot 1}{e^{11}})$

Factor $e^{2}$ out of $e^{11}$ .

$\ln (\frac{e^{2} \cdot 1}{e^{2} e^{9}})$

Cancel the common factor.

$\ln (\frac{e^{2} \cdot 1}{e^{2} e^{9}})$

Rewrite the expression.

$\ln (\frac{1}{e^{9}})$

Move $e^{9}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\ln (e^{- 9})$

Expand $\ln (e^{- 9})$ by moving $- 9$ outside the logarithm.

$- 9 \ln (e)$

The natural logarithm of $e$ is $1$ .

$- 9 \cdot 1$

Multiply $- 9$ by $1$ .

$- 9$