Finite Math Examples

$0.02 = e^{- \frac{t}{250}}$

Step 1

Rewrite the equation as $e^{- \frac{t}{250}} = 0.02$ .

$e^{- \frac{t}{250}} = 0.02$

Step 2

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

$\ln (e^{- \frac{t}{250}}) = \ln (0.02)$

Step 3

Expand the left side.

Tap for more steps...

Step 3.1

Expand $\ln (e^{- \frac{t}{250}})$ by moving $- \frac{t}{250}$ outside the logarithm.

$- \frac{t}{250} \ln (e) = \ln (0.02)$

Step 3.2

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

$- \frac{t}{250} \cdot 1 = \ln (0.02)$

Step 3.3

Multiply $- 1$ by $1$ .

$- \frac{t}{250} = \ln (0.02)$

Step 4

Multiply both sides of the equation by $- 250$ .

$- 250 (- \frac{t}{250}) = - 250 \ln (0.02)$

Step 5

Simplify the left side.

Tap for more steps...

Step 5.1

Simplify $- 250 (- \frac{t}{250})$ .

Tap for more steps...

Step 5.1.1

Cancel the common factor of $250$ .

Tap for more steps...

Step 5.1.1.1

Move the leading negative in $- \frac{t}{250}$ into the numerator.

$- 250 \frac{- t}{250} = - 250 \ln (0.02)$

Step 5.1.1.2

Factor $250$ out of $- 250$ .

$250 (- 1) \frac{- t}{250} = - 250 \ln (0.02)$

Step 5.1.1.3

Cancel the common factor.

$250 \cdot - 1 \frac{- t}{250} = - 250 \ln (0.02)$

Step 5.1.1.4

Rewrite the expression.

$- - t = - 250 \ln (0.02)$

Step 5.1.2

Multiply.

Tap for more steps...

Step 5.1.2.1

Multiply $- 1$ by $- 1$ .

$1 t = - 250 \ln (0.02)$

Step 5.1.2.2

Multiply $t$ by $1$ .

$t = - 250 \ln (0.02)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = - 250 \ln (0.02)$

Decimal Form:

$t = 978.00575135 \dots$