Solve for x natural log of x^2-1=3

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

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

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

Solve for $x$

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Exponentiation and log are inverse functions.

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

Add $1$ to both sides of the equation.

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

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

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

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

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Simplify the right side of the equation.

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

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

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 $b = 1$ .

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

Simplify.

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Multiply $- 1$ by $1$ .

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

One to any power is one.

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

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

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

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

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

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

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

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

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:

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

Solve for x natural log of x^2-1=3

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