Basic Math Examples

$15 = 157 e^{\frac{- t}{3700000}}$

Step 1

Rewrite the equation as $157 e^{\frac{- t}{3700000}} = 15$ .

$157 e^{\frac{- t}{3700000}} = 15$

Step 2

Divide each term in $157 e^{\frac{- t}{3700000}} = 15$ by $157$ and simplify.

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Step 2.1

Divide each term in $157 e^{\frac{- t}{3700000}} = 15$ by $157$ .

$\frac{157 e^{\frac{- t}{3700000}}}{157} = \frac{15}{157}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $157$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{157 e^{\frac{- t}{3700000}}}{157} = \frac{15}{157}$

Step 2.2.1.2

Divide $e^{\frac{- t}{3700000}}$ by $1$ .

$e^{\frac{- t}{3700000}} = \frac{15}{157}$

Step 2.2.2

Move the negative in front of the fraction.

$e^{- \frac{t}{3700000}} = \frac{15}{157}$

Step 3

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

$\ln (e^{- \frac{t}{3700000}}) = \ln (\frac{15}{157})$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{- \frac{t}{3700000}})$ by moving $- \frac{t}{3700000}$ outside the logarithm.

$- \frac{t}{3700000} \ln (e) = \ln (\frac{15}{157})$

Step 4.2

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

$- \frac{t}{3700000} \cdot 1 = \ln (\frac{15}{157})$

Step 4.3

Multiply $- 1$ by $1$ .

$- \frac{t}{3700000} = \ln (\frac{15}{157})$

Step 5

Multiply both sides of the equation by $- 3700000$ .

$- 3700000 (- \frac{t}{3700000}) = - 3700000 \ln (\frac{15}{157})$

Step 6

Simplify the left side.

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Step 6.1

Simplify $- 3700000 (- \frac{t}{3700000})$ .

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Step 6.1.1

Cancel the common factor of $3700000$ .

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Step 6.1.1.1

Move the leading negative in $- \frac{t}{3700000}$ into the numerator.

$- 3700000 \frac{- t}{3700000} = - 3700000 \ln (\frac{15}{157})$

Step 6.1.1.2

Factor $3700000$ out of $- 3700000$ .

$3700000 (- 1) \frac{- t}{3700000} = - 3700000 \ln (\frac{15}{157})$

Step 6.1.1.3

Cancel the common factor.

$3700000 \cdot - 1 \frac{- t}{3700000} = - 3700000 \ln (\frac{15}{157})$

Step 6.1.1.4

Rewrite the expression.

$- - t = - 3700000 \ln (\frac{15}{157})$

Step 6.1.2

Multiply.

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Step 6.1.2.1

Multiply $- 1$ by $- 1$ .

$1 t = - 3700000 \ln (\frac{15}{157})$

Step 6.1.2.2

Multiply $t$ by $1$ .

$t = - 3700000 \ln (\frac{15}{157})$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = - 3700000 \ln (\frac{15}{157})$

Decimal Form:

$t = 8688323.73571056$