Calculus Examples

$367.8 = 1000 e^{- \frac{x}{y}}$

Step 1

Rewrite the equation as $1000 e^{- \frac{x}{y}} = 367.8$ .

$1000 e^{- \frac{x}{y}} = 367.8$

Step 2

Divide each term in $1000 e^{- \frac{x}{y}} = 367.8$ by $1000$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $1000 e^{- \frac{x}{y}} = 367.8$ by $1000$ .

$\frac{1000 e^{- \frac{x}{y}}}{1000} = \frac{367.8}{1000}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $1000$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{1000 e^{- \frac{x}{y}}}{1000} = \frac{367.8}{1000}$

Step 2.2.1.2

Divide $e^{- \frac{x}{y}}$ by $1$ .

$e^{- \frac{x}{y}} = \frac{367.8}{1000}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Divide $367.8$ by $1000$ .

$e^{- \frac{x}{y}} = 0.3678$

Step 3

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

$\ln (e^{- \frac{x}{y}}) = \ln (0.3678)$

Step 4

Expand the left side.

Tap for more steps...

Step 4.1

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

$- \frac{x}{y} \ln (e) = \ln (0.3678)$

Step 4.2

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

$- \frac{x}{y} \cdot 1 = \ln (0.3678)$

Step 4.3

Multiply $- 1$ by $1$ .

$- \frac{x}{y} = \ln (0.3678)$

Step 5

Divide each term in $- \frac{x}{y} = \ln (0.3678)$ by $- 1$ and simplify.

Tap for more steps...

Step 5.1

Divide each term in $- \frac{x}{y} = \ln (0.3678)$ by $- 1$ .

$\frac{- \frac{x}{y}}{- 1} = \frac{\ln (0.3678)}{- 1}$

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{y}}{1} = \frac{\ln (0.3678)}{- 1}$

Step 5.2.2

Divide $\frac{x}{y}$ by $1$ .

$\frac{x}{y} = \frac{\ln (0.3678)}{- 1}$

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

Move the negative one from the denominator of $\frac{\ln (0.3678)}{- 1}$ .

$\frac{x}{y} = - 1 \cdot \ln (0.3678)$

Step 5.3.2

Rewrite $- 1 \cdot \ln (0.3678)$ as $- \ln (0.3678)$ .

$\frac{x}{y} = - \ln (0.3678)$

Step 6

Multiply both sides by $y$ .

$\frac{x}{y} y = - \ln (0.3678) y$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Simplify the left side.

Tap for more steps...

Step 7.1.1

Cancel the common factor of $y$ .

Tap for more steps...

Step 7.1.1.1

Cancel the common factor.

$\frac{x}{y} y = - \ln (0.3678) y$

Step 7.1.1.2

Rewrite the expression.

$x = - \ln (0.3678) y$

Step 7.2

Simplify the right side.

Tap for more steps...

Step 7.2.1

Reorder factors in $- \ln (0.3678) y$ .

$x = - y \ln (0.3678)$