Calculus Examples

$0.18 = 0.2 \cdot {0.5}^{\frac{t}{5730}}$

Step 1

Rewrite the equation as $0.2 \cdot {0.5}^{\frac{t}{5730}} = 0.18$ .

$0.2 \cdot {0.5}^{\frac{t}{5730}} = 0.18$

Step 2

Divide each term in $0.2 \cdot {0.5}^{\frac{t}{5730}} = 0.18$ by $0.2$ and simplify.

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Step 2.1

Divide each term in $0.2 \cdot {0.5}^{\frac{t}{5730}} = 0.18$ by $0.2$ .

$\frac{0.2 \cdot {0.5}^{\frac{t}{5730}}}{0.2} = \frac{0.18}{0.2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $0.2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{0.2 \cdot {0.5}^{\frac{t}{5730}}}{0.2} = \frac{0.18}{0.2}$

Step 2.2.1.2

Divide ${0.5}^{\frac{t}{5730}}$ by $1$ .

${0.5}^{\frac{t}{5730}} = \frac{0.18}{0.2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0.18$ by $0.2$ .

${0.5}^{\frac{t}{5730}} = 0.9$

Step 3

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

$\ln ({0.5}^{\frac{t}{5730}}) = \ln (0.9)$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln ({0.5}^{\frac{t}{5730}})$ by moving $\frac{t}{5730}$ outside the logarithm.

$\frac{t}{5730} \ln (0.5) = \ln (0.9)$

Step 4.2

Combine $\frac{t}{5730}$ and $\ln (0.5)$ .

$\frac{t \ln (0.5)}{5730} = \ln (0.9)$

Step 5

Multiply both sides of the equation by $\frac{5730}{\ln (0.5)}$ .

$\frac{5730}{\ln (0.5)} \cdot \frac{t \ln (0.5)}{5730} = \frac{5730}{\ln (0.5)} \ln (0.9)$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{5730}{\ln (0.5)} \cdot \frac{t \ln (0.5)}{5730}$ .

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Step 6.1.1.1

Cancel the common factor of $5730$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{5730}{\ln (0.5)} \cdot \frac{t \ln (0.5)}{5730} = \frac{5730}{\ln (0.5)} \ln (0.9)$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{\ln (0.5)} (t \ln (0.5)) = \frac{5730}{\ln (0.5)} \ln (0.9)$

Step 6.1.1.2

Cancel the common factor of $\ln (0.5)$ .

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Step 6.1.1.2.1

Factor $\ln (0.5)$ out of $t \ln (0.5)$ .

$\frac{1}{\ln (0.5)} (\ln (0.5) t) = \frac{5730}{\ln (0.5)} \ln (0.9)$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{\ln (0.5)} (\ln (0.5) t) = \frac{5730}{\ln (0.5)} \ln (0.9)$

Step 6.1.1.2.3

Rewrite the expression.

$t = \frac{5730}{\ln (0.5)} \ln (0.9)$

Step 6.2

Simplify the right side.

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Step 6.2.1

Combine $\frac{5730}{\ln (0.5)}$ and $\ln (0.9)$ .

$t = \frac{5730 \ln (0.9)}{\ln (0.5)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = \frac{5730 \ln (0.9)}{\ln (0.5)}$

Decimal Form:

$t = 870.97772544 \dots$