Precalculus Examples

$\ln (\frac{1}{e}) = x$

Step 1

Rewrite the equation as $x = \ln (\frac{1}{e})$ .

$x = \ln (\frac{1}{e})$

Step 2

Simplify $\ln (\frac{1}{e})$ .

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Step 2.1

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

$x = \ln (1 e^{- 1})$

Step 2.2

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

$x = \ln (1) + \ln (e^{- 1})$

Step 2.3

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

$x = \ln (1) - \ln (e)$

Step 2.4

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

$x = \ln (1) - 1 \cdot 1$

Step 2.5

Multiply $- 1$ by $1$ .

$x = \ln (1) - 1$

Step 2.6

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

$x = 0 - 1$

Step 2.7

Subtract $1$ from $0$ .

$x = - 1$