Algebra Examples

$y = e^{- x^{2}}$

Step 1

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

$e^{- x^{2}} = y$

Step 2

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

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

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.

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

Step 3.2

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

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

Step 3.3

Multiply $- 1$ by $1$ .

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

Step 4

Divide each term in $- x^{2} = \ln (y)$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- x^{2} = \ln (y)$ by $- 1$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

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

Step 4.2.2

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

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative one from the denominator of $\frac{\ln (y)}{- 1}$ .

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

Step 4.3.2

Rewrite $- 1 \cdot \ln (y)$ as $- \ln (y)$ .

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

Step 5

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

$x = \pm \sqrt{- \ln (y)}$

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

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

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

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