Precalculus Examples

$\ln (\frac{1}{e^{4}}) + \ln (4 e^{2})$

Step 1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (\frac{1}{e^{4}} (4 e^{2}))$

Step 2

Cancel the common factor of $e^{2}$ .

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Step 2.1

Factor $e^{2}$ out of $e^{4}$ .

$\ln (\frac{1}{e^{2} e^{2}} (4 e^{2}))$

Step 2.2

Factor $e^{2}$ out of $4 e^{2}$ .

$\ln (\frac{1}{e^{2} e^{2}} (e^{2} \cdot 4))$

Step 2.3

Cancel the common factor.

$\ln (\frac{1}{e^{2} e^{2}} (e^{2} \cdot 4))$

Step 2.4

Rewrite the expression.

$\ln (\frac{1}{e^{2}} \cdot 4)$

Step 3

Combine $\frac{1}{e^{2}}$ and $4$ .

$\ln (\frac{4}{e^{2}})$

Step 4

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

$\ln (4 e^{- 2})$

Step 5

Rewrite $\ln (4 e^{- 2})$ as $\ln (4) + \ln (e^{- 2})$ .

$\ln (4) + \ln (e^{- 2})$

Step 6

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

$\ln (4) - 2 \ln (e)$

Step 7

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

$\ln (4) - 2 \cdot 1$

Step 8

Multiply $- 2$ by $1$ .

$\ln (4) - 2$

Step 9

Simplify each term.

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Step 9.1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

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

Step 9.2

Expand $\ln (2^{2})$ by moving $2$ outside the logarithm.

$2 \ln (2) - 2$

Step 10

The result can be shown in multiple forms.

Exact Form:

$2 \ln (2) - 2$

Decimal Form:

$- 0.61370563 \dots$