Calculus Examples

$\ln (e^{x^{2}}) = 25$

Step 1

Expand $\ln (e^{x^{2}})$ .

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Step 1.1

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

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

Step 1.2

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

$x^{2} \cdot 1$

Step 1.3

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

$x^{2}$

Step 2

The expanded equation is $x^{2} = 25$ .

$x^{2} = 25$

Step 3

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

$x = \pm \sqrt{25}$

Step 4

Simplify $\pm \sqrt{25}$ .

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Step 4.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2}}$

Step 4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 5$

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.

$x = 5$

Step 5.2

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

$x = - 5$

Step 5.3

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

$x = 5, - 5$