Calculus Examples

$e^{0.05 x} = 2$

Step 1

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

$\ln (e^{0.05 x}) = \ln (2)$

Step 2

Expand the left side.

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Step 2.1

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

$0.05 x \ln (e) = \ln (2)$

Step 2.2

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

$0.05 x \cdot 1 = \ln (2)$

Step 2.3

Multiply $0.05$ by $1$ .

$0.05 x = \ln (2)$

Step 3

Divide each term in $0.05 x = \ln (2)$ by $0.05$ and simplify.

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Step 3.1

Divide each term in $0.05 x = \ln (2)$ by $0.05$ .

$\frac{0.05 x}{0.05} = \frac{\ln (2)}{0.05}$

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.

$\frac{0.05 x}{0.05} = \frac{\ln (2)}{0.05}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (2)}{0.05}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Replace $e$ with an approximation.

$x = \frac{\log_{2.71828182} (2)}{0.05}$

Step 3.3.2

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

$x = \frac{0.69314718}{0.05}$

Step 3.3.3

Divide $0.69314718$ by $0.05$ .

$x = 13.86294361$