Precalculus Examples

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

Step 1

Simplify $\ln (e^{x})$ .

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Step 1.1

Use logarithm rules to move $x$ out of the exponent.

$x \ln (e) = x$

Step 1.2

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

$x \cdot 1 = x$

Step 1.3

Multiply $x$ by $1$ .

$x = x$

Step 2

Move all terms containing $x$ to the left side of the equation.

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Step 2.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 2.2

Subtract $x$ from $x$ .

$0 = 0$

Step 3

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$