Calculus Examples

$7 - e^{0.5 t} = 3$

Step 1

Move all terms not containing $t$ to the right side of the equation.

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Step 1.1

Subtract $7$ from both sides of the equation.

$- e^{0.5 t} = 3 - 7$

Step 1.2

Subtract $7$ from $3$ .

$- e^{0.5 t} = - 4$

Step 2

Divide each term in $- e^{0.5 t} = - 4$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- e^{0.5 t} = - 4$ by $- 1$ .

$\frac{- e^{0.5 t}}{- 1} = \frac{- 4}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{e^{0.5 t}}{1} = \frac{- 4}{- 1}$

Step 2.2.2

Divide $e^{0.5 t}$ by $1$ .

$e^{0.5 t} = \frac{- 4}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 4$ by $- 1$ .

$e^{0.5 t} = 4$

Step 3

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

$\ln (e^{0.5 t}) = \ln (4)$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{0.5 t})$ by moving $0.5 t$ outside the logarithm.

$0.5 t \ln (e) = \ln (4)$

Step 4.2

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

$0.5 t \cdot 1 = \ln (4)$

Step 4.3

Multiply $0.5$ by $1$ .

$0.5 t = \ln (4)$

Step 5

Divide each term in $0.5 t = \ln (4)$ by $0.5$ and simplify.

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Step 5.1

Divide each term in $0.5 t = \ln (4)$ by $0.5$ .

$\frac{0.5 t}{0.5} = \frac{\ln (4)}{0.5}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $0.5$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{0.5 t}{0.5} = \frac{\ln (4)}{0.5}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (4)}{0.5}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (4)}{0.5}$

Step 5.3.2

Log base $2.71828182$ of $4$ is approximately $1.38629436$ .

$t = \frac{1.38629436}{0.5}$

Step 5.3.3

Divide $1.38629436$ by $0.5$ .

$t = 2.77258872$