Calculus Examples

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

Step 1

Simplify the left side.

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Step 1.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

$\ln (x \cdot x) = 2$

Step 1.3

Multiply $x$ by $x$ .

$\ln (x^{2}) = 2$

Step 2

Write in exponential form.

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Step 2.1

For logarithmic equations, $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ such that $x > 0$ , $b > 0$ , and $b \neq 1$ . In this case, $b = e$ , $x = x^{2}$ , and $y = 2$ .

$b = e$

$x = x^{2}$

$y = 2$

Step 2.2

Substitute the values of $b$ , $x$ , and $y$ into the equation $b^{y} = x$ .

$e^{2} = x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

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

$x^{2} = e^{2}$

Step 3.2

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| x | = | e |$

Step 3.3

Solve for $x$ .

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Step 3.3.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x = \pm | e |$

Step 3.3.2

$e$ is approximately $2.71828182$ which is positive so remove the absolute value

$x = \pm e$

Step 3.3.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = e$

Step 3.3.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - e$

Step 3.3.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = e, - e$

Step 4

Exclude the solutions that do not make $\ln (x) - \ln (\frac{1}{x}) = 2$ true.

$x = e$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = e$

Decimal Form:

$x = 2.71828182 \dots$