Calculus Examples

$e^{- 0.07 t} = 0.05$

Step 1

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

$\ln (e^{- 0.07 t}) = \ln (0.05)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (e^{- 0.07 t})$ by moving $- 0.07 t$ outside the logarithm.

$- 0.07 t \ln (e) = \ln (0.05)$

Step 2.2

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

$- 0.07 t \cdot 1 = \ln (0.05)$

Step 2.3

Multiply $- 0.07$ by $1$ .

$- 0.07 t = \ln (0.05)$

Step 3

Divide each term in $- 0.07 t = \ln (0.05)$ by $- 0.07$ and simplify.

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Step 3.1

Divide each term in $- 0.07 t = \ln (0.05)$ by $- 0.07$ .

$\frac{- 0.07 t}{- 0.07} = \frac{\ln (0.05)}{- 0.07}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 0.07$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 0.07 t}{- 0.07} = \frac{\ln (0.05)}{- 0.07}$

Step 3.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (0.05)}{- 0.07}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (0.05)}{0.07}$

Step 3.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (0.05)}{0.07}$

Step 3.3.3

Log base $2.71828182$ of $0.05$ is approximately $- 2.99573227$ .

$t = - \frac{- 2.99573227}{0.07}$

Step 3.3.4

Divide $- 2.99573227$ by $0.07$ .

$t = - - 42.79617533$

Step 3.3.5

Multiply $- 1$ by $- 42.79617533$ .

$t = 42.79617533$