Evaluate natural log of e^3- natural log of e^13

$\ln (e^{3}) - \ln (e^{13})$

Step 1

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

$\ln (\frac{e^{3}}{e^{13}})$

Step 2

Reduce the expression $\frac{e^{3}}{e^{13}}$ by cancelling the common factors.

Multiply by $1$ .

$\ln (\frac{e^{3} \cdot 1}{e^{13}})$

Factor $e^{3}$ out of $e^{13}$ .

$\ln (\frac{e^{3} \cdot 1}{e^{3} e^{10}})$

Cancel the common factor.

$\ln (\frac{e^{3} \cdot 1}{e^{3} e^{10}})$

Rewrite the expression.

$\ln (\frac{1}{e^{10}})$

Step 3

Move $e^{10}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\ln (e^{- 10})$

Step 4

Expand $\ln (e^{- 10})$ by moving $- 10$ outside the logarithm.

$- 10 \ln (e)$

Step 5

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

$- 10 \cdot 1$

Step 6

Multiply $- 10$ by $1$ .

$- 10$