Calculus Examples

$e^{0.05 t} = 3$

Step 1

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

$\ln (e^{0.05 t}) = \ln (3)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (e^{0.05 t})$ by moving $0.05 t$ outside the logarithm.

$0.05 t \ln (e) = \ln (3)$

Step 2.2

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

$0.05 t \cdot 1 = \ln (3)$

Step 2.3

Multiply $0.05$ by $1$ .

$0.05 t = \ln (3)$

Step 3

Divide each term in $0.05 t = \ln (3)$ by $0.05$ and simplify.

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Step 3.1

Divide each term in $0.05 t = \ln (3)$ by $0.05$ .

$\frac{0.05 t}{0.05} = \frac{\ln (3)}{0.05}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $0.05$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{0.05 t}{0.05} = \frac{\ln (3)}{0.05}$

Step 3.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (3)}{0.05}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (3)}{0.05}$

Step 3.3.2

Log base $2.71828182$ of $3$ is approximately $1.09861228$ .

$t = \frac{1.09861228}{0.05}$

Step 3.3.3

Divide $1.09861228$ by $0.05$ .

$t = 21.97224577$