Evaluate natural log of e^5- natural log of e^16

$\ln (e^{5}) - \ln (e^{16})$

Step 1

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

$\ln (\frac{e^{5}}{e^{16}})$

Step 2

Reduce the expression $\frac{e^{5}}{e^{16}}$ by cancelling the common factors.

Multiply by $1$ .

$\ln (\frac{e^{5} \cdot 1}{e^{16}})$

Factor $e^{5}$ out of $e^{16}$ .

$\ln (\frac{e^{5} \cdot 1}{e^{5} e^{11}})$

Cancel the common factor.

$\ln (\frac{e^{5} \cdot 1}{e^{5} e^{11}})$

Rewrite the expression.

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

Step 3

Move $e^{11}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 4

Expand $\ln (e^{- 11})$ by moving $- 11$ outside the logarithm.

$- 11 \ln (e)$

Step 5

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

$- 11 \cdot 1$

Step 6

Multiply $- 11$ by $1$ .

$- 11$