Calculus Examples

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

Step 1

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

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

Step 2

Rewrite $\ln (x^{2} - 1) = 3$ 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 $b^{y} = x$ .

$e^{3} = x^{2} - 1$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $x^{2} - 1 = e^{3}$ .

$x^{2} - 1 = e^{3}$

Step 3.2

Add $1$ to both sides of the equation.

$x^{2} = e^{3} + 1$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{e^{3} + 1}$

Step 3.4

Simplify $\pm \sqrt{e^{3} + 1}$ .

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Step 3.4.1

Rewrite $1$ as $1^{3}$ .

$x = \pm \sqrt{e^{3} + 1^{3}}$

Step 3.4.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = e$ and $b = 1$ .

$x = \pm \sqrt{(e + 1) (e^{2} - e \cdot 1 + 1^{2})}$

Step 3.4.3

Simplify.

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Step 3.4.3.1

Multiply $- 1$ by $1$ .

$x = \pm \sqrt{(e + 1) (e^{2} - e + 1^{2})}$

Step 3.4.3.2

One to any power is one.

$x = \pm \sqrt{(e + 1) (e^{2} - e + 1)}$

Step 3.5

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

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Step 3.5.1

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

$x = \sqrt{(e + 1) (e^{2} - e + 1)}$

Step 3.5.2

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

$x = - \sqrt{(e + 1) (e^{2} - e + 1)}$

Step 3.5.3

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

$x = \sqrt{(e + 1) (e^{2} - e + 1)}, - \sqrt{(e + 1) (e^{2} - e + 1)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{(e + 1) (e^{2} - e + 1)}, - \sqrt{(e + 1) (e^{2} - e + 1)}$

Decimal Form:

$x = 4.59189905 \dots, - 4.59189905 \dots$