Precalculus Examples

$1249 = 175 e^{- 0.04 t}$

Step 1

Rewrite the equation as $175 e^{- 0.04 t} = 1249$ .

$175 e^{- 0.04 t} = 1249$

Step 2

Divide each term in $175 e^{- 0.04 t} = 1249$ by $175$ and simplify.

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Step 2.1

Divide each term in $175 e^{- 0.04 t} = 1249$ by $175$ .

$\frac{175 e^{- 0.04 t}}{175} = \frac{1249}{175}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $175$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{175 e^{- 0.04 t}}{175} = \frac{1249}{175}$

Step 2.2.1.2

Divide $e^{- 0.04 t}$ by $1$ .

$e^{- 0.04 t} = \frac{1249}{175}$

Step 3

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

$\ln (e^{- 0.04 t}) = \ln (\frac{1249}{175})$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{- 0.04 t})$ by moving $- 0.04 t$ outside the logarithm.

$- 0.04 t \ln (e) = \ln (\frac{1249}{175})$

Step 4.2

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

$- 0.04 t \cdot 1 = \ln (\frac{1249}{175})$

Step 4.3

Multiply $- 0.04$ by $1$ .

$- 0.04 t = \ln (\frac{1249}{175})$

Step 5

Divide each term in $- 0.04 t = \ln (\frac{1249}{175})$ by $- 0.04$ and simplify.

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Step 5.1

Divide each term in $- 0.04 t = \ln (\frac{1249}{175})$ by $- 0.04$ .

$\frac{- 0.04 t}{- 0.04} = \frac{\ln (\frac{1249}{175})}{- 0.04}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 0.04$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 0.04 t}{- 0.04} = \frac{\ln (\frac{1249}{175})}{- 0.04}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{1249}{175})}{- 0.04}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (\frac{1249}{175})}{0.04}$

Step 5.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (\frac{1249}{175})}{0.04}$

Step 5.3.3

Divide $1249$ by $175$ .

$t = - \frac{\log_{2.71828182} (7.13714285)}{0.04}$

Step 5.3.4

Log base $2.71828182$ of $7.13714285$ is approximately $1.96531253$ .

$t = - \frac{1.96531253}{0.04}$

Step 5.3.5

Divide $1.96531253$ by $0.04$ .

$t = - 1 \cdot 49.1328134$

Step 5.3.6

Multiply $- 1$ by $49.1328134$ .

$t = - 49.1328134$