Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Expand $\ln (e^{4})$ by moving $4$ outside the logarithm.

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

Step 1.2

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

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

Step 1.3

Multiply $4$ by $1$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $\frac{1}{e}$ as $1 e^{- 1}$ .

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

Step 2.2

Rewrite $\ln (1 e^{- 1})$ as $\ln (1) + \ln (e^{- 1})$ .

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

Step 2.3

Use logarithm rules to move $- 1$ out of the exponent.

$\frac{4}{\ln (1) - \ln (e)}$

Step 2.4

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

$\frac{4}{\ln (1) - 1 \cdot 1}$

Step 2.5

Multiply $- 1$ by $1$ .

$\frac{4}{\ln (1) - 1}$

Step 2.6

The natural logarithm of $1$ is $0$ .

$\frac{4}{0 - 1}$

Step 2.7

Subtract $1$ from $0$ .

$\frac{4}{- 1}$

Step 3

Divide $4$ by $- 1$ .

$- 4$