Precalculus Examples

$2 x \ln (\frac{1}{x}) - x = 0$

Step 1

Add $x$ to both sides of the equation.

$2 x \ln (\frac{1}{x}) = x$

Step 2

Divide each term in $2 x \ln (\frac{1}{x}) = x$ by $2 x$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $2 x \ln (\frac{1}{x}) = x$ by $2 x$ .

$\frac{2 x \ln (\frac{1}{x})}{2 x} = \frac{x}{2 x}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{2 x \ln (\frac{1}{x})}{2 x} = \frac{x}{2 x}$

Step 2.2.1.2

Rewrite the expression.

$\frac{x \ln (\frac{1}{x})}{x} = \frac{x}{2 x}$

Step 2.2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.2.2.1

Cancel the common factor.

$\frac{x \ln (\frac{1}{x})}{x} = \frac{x}{2 x}$

Step 2.2.2.2

Divide $\ln (\frac{1}{x})$ by $1$ .

$\ln (\frac{1}{x}) = \frac{x}{2 x}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.3.1.1

Cancel the common factor.

$\ln (\frac{1}{x}) = \frac{x}{2 x}$

Step 2.3.1.2

Rewrite the expression.

$\ln (\frac{1}{x}) = \frac{1}{2}$

Step 3

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

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

Step 4

Rewrite $\ln (\frac{1}{x}) = \frac{1}{2}$ 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 $b^{y} = x$ .

$e^{\frac{1}{2}} = \frac{1}{x}$

Step 5

Solve for $x$ .

Tap for more steps...

Step 5.1

Rewrite the equation as $\frac{1}{x} = e^{\frac{1}{2}}$ .

$\frac{1}{x} = e^{\frac{1}{2}}$

Step 5.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 5.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

$x$

Step 5.3

Multiply each term in $\frac{1}{x} = e^{\frac{1}{2}}$ by $x$ to eliminate the fractions.

Tap for more steps...

Step 5.3.1

Multiply each term in $\frac{1}{x} = e^{\frac{1}{2}}$ by $x$ .

$\frac{1}{x} x = e^{\frac{1}{2}} x$

Step 5.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.3.2.1.1

Cancel the common factor.

$\frac{1}{x} x = e^{\frac{1}{2}} x$

Step 5.3.2.1.2

Rewrite the expression.

$1 = e^{\frac{1}{2}} x$

Step 5.3.3

Simplify the right side.

Tap for more steps...

Step 5.3.3.1

Reorder factors in $e^{\frac{1}{2}} x$ .

$1 = x e^{\frac{1}{2}}$

Step 5.4

Solve the equation.

Tap for more steps...

Step 5.4.1

Rewrite the equation as $x e^{\frac{1}{2}} = 1$ .

$x e^{\frac{1}{2}} = 1$

Step 5.4.2

Divide each term in $x e^{\frac{1}{2}} = 1$ by $e^{\frac{1}{2}}$ and simplify.

Tap for more steps...

Step 5.4.2.1

Divide each term in $x e^{\frac{1}{2}} = 1$ by $e^{\frac{1}{2}}$ .

$\frac{x e^{\frac{1}{2}}}{e^{\frac{1}{2}}} = \frac{1}{e^{\frac{1}{2}}}$

Step 5.4.2.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.2.1

Cancel the common factor.

$\frac{x e^{\frac{1}{2}}}{e^{\frac{1}{2}}} = \frac{1}{e^{\frac{1}{2}}}$

Step 5.4.2.2.2

Divide $x$ by $1$ .

$x = \frac{1}{e^{\frac{1}{2}}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{e^{\frac{1}{2}}}$

Decimal Form:

$x = 0.60653065 \dots$