Precalculus Examples

$x \ln (x) = x$

Step 1

Subtract $x$ from both sides of the equation.

$x \ln (x) - x = 0$

Step 2

Add $x$ to both sides of the equation.

$x \ln (x) = x$

Step 3

Divide each term in $x \ln (x) = x$ by $x$ and simplify.

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Step 3.1

Divide each term in $x \ln (x) = x$ by $x$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Divide $\ln (x)$ by $1$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $x$ .

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

$\ln (x) = 1$

Step 4

To solve for $x$ , rewrite the equation using properties of logarithms.

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

Step 5

Rewrite $\ln (x) = 1$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{1} = x$

Step 6

Rewrite the equation as $x = e^{1}$ .

$x = e$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = e$

Decimal Form:

$x = 2.71828182 \dots$