Precalculus Examples

$150 = 100 e^{0.05 t}$

Step 1

Rewrite the equation as $100 e^{0.05 t} = 150$ .

$100 e^{0.05 t} = 150$

Step 2

Divide each term in $100 e^{0.05 t} = 150$ by $100$ and simplify.

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Step 2.1

Divide each term in $100 e^{0.05 t} = 150$ by $100$ .

$\frac{100 e^{0.05 t}}{100} = \frac{150}{100}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $100$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{100 e^{0.05 t}}{100} = \frac{150}{100}$

Step 2.2.1.2

Divide $e^{0.05 t}$ by $1$ .

$e^{0.05 t} = \frac{150}{100}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $150$ and $100$ .

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Step 2.3.1.1

Factor $50$ out of $150$ .

$e^{0.05 t} = \frac{50 (3)}{100}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $50$ out of $100$ .

$e^{0.05 t} = \frac{50 \cdot 3}{50 \cdot 2}$

Step 2.3.1.2.2

Cancel the common factor.

$e^{0.05 t} = \frac{50 \cdot 3}{50 \cdot 2}$

Step 2.3.1.2.3

Rewrite the expression.

$e^{0.05 t} = \frac{3}{2}$

Step 3

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

$\ln (e^{0.05 t}) = \ln (\frac{3}{2})$

Step 4

Expand the left side.

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Step 4.1

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

$0.05 t \ln (e) = \ln (\frac{3}{2})$

Step 4.2

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

$0.05 t \cdot 1 = \ln (\frac{3}{2})$

Step 4.3

Multiply $0.05$ by $1$ .

$0.05 t = \ln (\frac{3}{2})$

Step 5

Divide each term in $0.05 t = \ln (\frac{3}{2})$ by $0.05$ and simplify.

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Step 5.1

Divide each term in $0.05 t = \ln (\frac{3}{2})$ by $0.05$ .

$\frac{0.05 t}{0.05} = \frac{\ln (\frac{3}{2})}{0.05}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $0.05$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{0.05 t}{0.05} = \frac{\ln (\frac{3}{2})}{0.05}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{3}{2})}{0.05}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (\frac{3}{2})}{0.05}$

Step 5.3.2

Divide $3$ by $2$ .

$t = \frac{\log_{2.71828182} (1.5)}{0.05}$

Step 5.3.3

Log base $2.71828182$ of $1.5$ is approximately $0.4054651$ .

$t = \frac{0.4054651}{0.05}$

Step 5.3.4

Divide $0.4054651$ by $0.05$ .

$t = 8.10930216$