Calculus Examples

$2 = e^{0.03 t}$

Step 1

Rewrite the equation as $e^{0.03 t} = 2$ .

$e^{0.03 t} = 2$

Step 2

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

$\ln (e^{0.03 t}) = \ln (2)$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{0.03 t})$ by moving $0.03 t$ outside the logarithm.

$0.03 t \ln (e) = \ln (2)$

Step 3.2

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

$0.03 t \cdot 1 = \ln (2)$

Step 3.3

Multiply $0.03$ by $1$ .

$0.03 t = \ln (2)$

Step 4

Divide each term in $0.03 t = \ln (2)$ by $0.03$ and simplify.

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Step 4.1

Divide each term in $0.03 t = \ln (2)$ by $0.03$ .

$\frac{0.03 t}{0.03} = \frac{\ln (2)}{0.03}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $0.03$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{0.03 t}{0.03} = \frac{\ln (2)}{0.03}$

Step 4.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (2)}{0.03}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (2)}{0.03}$

Step 4.3.2

Log base $2.71828182$ of $2$ is approximately $0.69314718$ .

$t = \frac{0.69314718}{0.03}$

Step 4.3.3

Divide $0.69314718$ by $0.03$ .

$t = 23.10490601$