Precalculus Examples

$\ln (3 e^{3}) - \ln (e^{5})$

Step 1

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

$\ln (\frac{3 e^{3}}{e^{5}})$

Step 2

Reduce the expression $\frac{3 e^{3}}{e^{5}}$ by cancelling the common factors.

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Step 2.1

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

$\ln (\frac{e^{3} \cdot 3}{e^{5}})$

Step 2.2

Factor $e^{3}$ out of $e^{5}$ .

$\ln (\frac{e^{3} \cdot 3}{e^{3} e^{2}})$

Step 2.3

Cancel the common factor.

$\ln (\frac{e^{3} \cdot 3}{e^{3} e^{2}})$

Step 2.4

Rewrite the expression.

$\ln (\frac{3}{e^{2}})$

Step 3

Move $e^{2}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\ln (3 e^{- 2})$

Step 4

Rewrite $\ln (3 e^{- 2})$ as $\ln (3) + \ln (e^{- 2})$ .

$\ln (3) + \ln (e^{- 2})$

Step 5

Use logarithm rules to move $- 2$ out of the exponent.

$\ln (3) - 2 \ln (e)$

Step 6

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

$\ln (3) - 2 \cdot 1$

Step 7

Multiply $- 2$ by $1$ .

$\ln (3) - 2$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\ln (3) - 2$

Decimal Form:

$- 0.90138771 \dots$