Calculus Examples

$2 \ln (x) + \ln (\frac{1}{x}) = 3$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$2 \ln (x) + \ln (\frac{1}{x}) = 3$

Step 2

Simplify the left side.

Tap for more steps...

Step 2.1

Simplify $2 \ln (x) + \ln (\frac{1}{x})$ .

Tap for more steps...

Step 2.1.1

Simplify $2 \ln (x)$ by moving $2$ inside the logarithm.

$\ln (x^{2}) + \ln (\frac{1}{x}) = 3$

Step 2.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (x^{2} \frac{1}{x}) = 3$

Step 2.1.3

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.1.3.1

Factor $x$ out of $x^{2}$ .

$\ln (x \cdot x \frac{1}{x}) = 3$

Step 2.1.3.2

Cancel the common factor.

$\ln (x \cdot x \frac{1}{x}) = 3$

Step 2.1.3.3

Rewrite the expression.

$\ln (x) = 3$

Step 3

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

$e^{\ln (x)} = e^{3}$

Step 4

Rewrite $\ln (x) = 3$ 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 $b^{y} = x$ .

$e^{3} = x$

Step 5

Rewrite the equation as $x = e^{3}$ .

$x = e^{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = e^{3}$

Decimal Form:

$x = 20.08553692 \dots$