Finite Math Examples

$e^{5 x^{2}} = 6$

Step 1

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

$\ln (e^{5 x^{2}}) = \ln (6)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (e^{5 x^{2}})$ by moving $5 x^{2}$ outside the logarithm.

$5 x^{2} \ln (e) = \ln (6)$

Step 2.2

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

$5 x^{2} \cdot 1 = \ln (6)$

Step 2.3

Multiply $5$ by $1$ .

$5 x^{2} = \ln (6)$

Step 3

Divide each term in $5 x^{2} = \ln (6)$ by $5$ and simplify.

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Step 3.1

Divide each term in $5 x^{2} = \ln (6)$ by $5$ .

$\frac{5 x^{2}}{5} = \frac{\ln (6)}{5}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $5$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{5 x^{2}}{5} = \frac{\ln (6)}{5}$

Step 3.2.1.2

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

$x^{2} = \frac{\ln (6)}{5}$

Step 4

Rewrite $\frac{\ln (6)}{5}$ as $\frac{1}{5} \ln (6)$ .

$x^{2} = \frac{1}{5} \ln (6)$

Step 5

Simplify $\frac{1}{5} \ln (6)$ by moving $\frac{1}{5}$ inside the logarithm.

$x^{2} = \ln (6^{\frac{1}{5}})$

Step 6

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

$x = \pm \sqrt{\ln (6^{\frac{1}{5}})}$

Step 7

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

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Step 7.1

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

$x = \sqrt{\ln (6^{\frac{1}{5}})}$

Step 7.2

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

$x = - \sqrt{\ln (6^{\frac{1}{5}})}$

Step 7.3

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

$x = \sqrt{\ln (6^{\frac{1}{5}})}, - \sqrt{\ln (6^{\frac{1}{5}})}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{\ln (6^{\frac{1}{5}})}, - \sqrt{\ln (6^{\frac{1}{5}})}$

Decimal Form:

$x = 0.59862500 \dots, - 0.59862500 \dots$