Calculus Examples

$\frac{1}{2} \cdot e^{- 4 t} = 0.9$

Step 1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot e^{- 4 t}) = 2 \cdot 0.9$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $2 (\frac{1}{2} \cdot e^{- 4 t})$ .

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Step 2.1.1.1

Combine $\frac{1}{2}$ and $e^{- 4 t}$ .

$2 \frac{e^{- 4 t}}{2} = 2 \cdot 0.9$

Step 2.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.2.1

Cancel the common factor.

$2 \frac{e^{- 4 t}}{2} = 2 \cdot 0.9$

Step 2.1.1.2.2

Rewrite the expression.

$e^{- 4 t} = 2 \cdot 0.9$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $2$ by $0.9$ .

$e^{- 4 t} = 1.8$

Step 3

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

$\ln (e^{- 4 t}) = \ln (1.8)$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{- 4 t})$ by moving $- 4 t$ outside the logarithm.

$- 4 t \ln (e) = \ln (1.8)$

Step 4.2

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

$- 4 t \cdot 1 = \ln (1.8)$

Step 4.3

Multiply $- 4$ by $1$ .

$- 4 t = \ln (1.8)$

Step 5

Divide each term in $- 4 t = \ln (1.8)$ by $- 4$ and simplify.

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Step 5.1

Divide each term in $- 4 t = \ln (1.8)$ by $- 4$ .

$\frac{- 4 t}{- 4} = \frac{\ln (1.8)}{- 4}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 4$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 4 t}{- 4} = \frac{\ln (1.8)}{- 4}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (1.8)}{- 4}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (1.8)}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = - \frac{\ln (1.8)}{4}$

Decimal Form:

$t = - 0.14694666 \dots$