Precalculus Examples

$9 = e^{x^{2}}$

Step 1

Rewrite the equation as $e^{x^{2}} = 9$ .

$e^{x^{2}} = 9$

Step 2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x^{2}}) = \ln (9)$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{x^{2}})$ by moving $x^{2}$ outside the logarithm.

$x^{2} \ln (e) = \ln (9)$

Step 3.2

The natural logarithm of $e$ is $1$ .

$x^{2} \cdot 1 = \ln (9)$

Step 3.3

Multiply $x^{2}$ by $1$ .

$x^{2} = \ln (9)$

Step 4

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

$x = \pm \sqrt{\ln (9)}$

Step 5

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

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Step 5.1

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

$x = \sqrt{\ln (9)}$

Step 5.2

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

$x = - \sqrt{\ln (9)}$

Step 5.3

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

$x = \sqrt{\ln (9)}, - \sqrt{\ln (9)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{\ln (9)}, - \sqrt{\ln (9)}$

Decimal Form:

$x = 1.48230380 \dots, - 1.48230380 \dots$