Basic Math Examples

$65 = 200 {(\frac{1}{2})}^{\frac{- t}{180}}$

Step 1

Rewrite the equation as $200 {(\frac{1}{2})}^{\frac{- t}{180}} = 65$ .

$200 {(\frac{1}{2})}^{\frac{- t}{180}} = 65$

Step 2

Divide each term in $200 {(\frac{1}{2})}^{\frac{- t}{180}} = 65$ by $200$ and simplify.

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Step 2.1

Divide each term in $200 {(\frac{1}{2})}^{\frac{- t}{180}} = 65$ by $200$ .

$\frac{200 {(\frac{1}{2})}^{\frac{- t}{180}}}{200} = \frac{65}{200}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $200$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{200 {(\frac{1}{2})}^{\frac{- t}{180}}}{200} = \frac{65}{200}$

Step 2.2.1.2

Divide ${(\frac{1}{2})}^{\frac{- t}{180}}$ by $1$ .

${(\frac{1}{2})}^{\frac{- t}{180}} = \frac{65}{200}$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Move the negative in front of the fraction.

${(\frac{1}{2})}^{- \frac{t}{180}} = \frac{65}{200}$

Step 2.2.2.2

Apply the product rule to $\frac{1}{2}$ .

$\frac{1^{- \frac{t}{180}}}{2^{- \frac{t}{180}}} = \frac{65}{200}$

Step 2.2.2.3

One to any power is one.

$\frac{1}{2^{- \frac{t}{180}}} = \frac{65}{200}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $65$ and $200$ .

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Step 2.3.1.1

Factor $5$ out of $65$ .

$\frac{1}{2^{- \frac{t}{180}}} = \frac{5 (13)}{200}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $5$ out of $200$ .

$\frac{1}{2^{- \frac{t}{180}}} = \frac{5 \cdot 13}{5 \cdot 40}$

Step 2.3.1.2.2

Cancel the common factor.

$\frac{1}{2^{- \frac{t}{180}}} = \frac{5 \cdot 13}{5 \cdot 40}$

Step 2.3.1.2.3

Rewrite the expression.

$\frac{1}{2^{- \frac{t}{180}}} = \frac{13}{40}$

Step 3

Multiply both sides by $2^{- \frac{t}{180}}$ .

$\frac{1}{2^{- \frac{t}{180}}} \cdot 2^{- \frac{t}{180}} = \frac{13}{40} \cdot 2^{- \frac{t}{180}}$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $2^{- \frac{t}{180}}$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{1}{2^{- \frac{t}{180}}} \cdot 2^{- \frac{t}{180}} = \frac{13}{40} \cdot 2^{- \frac{t}{180}}$

Step 4.1.1.2

Rewrite the expression.

$1 = \frac{13}{40} \cdot 2^{- \frac{t}{180}}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Combine $\frac{13}{40}$ and $2^{- \frac{t}{180}}$ .

$1 = \frac{13 \cdot 2^{- \frac{t}{180}}}{40}$

Step 5

Solve for $t$ .

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Step 5.1

Rewrite the equation as $\frac{13 \cdot 2^{- \frac{t}{180}}}{40} = 1$ .

$\frac{13 \cdot 2^{- \frac{t}{180}}}{40} = 1$

Step 5.2

Multiply both sides by $40$ .

$\frac{13 \cdot 2^{- \frac{t}{180}}}{40} \cdot 40 = 1 \cdot 40$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Cancel the common factor of $40$ .

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Step 5.3.1.1.1

Cancel the common factor.

$\frac{13 \cdot 2^{- \frac{t}{180}}}{40} \cdot 40 = 1 \cdot 40$

Step 5.3.1.1.2

Rewrite the expression.

$13 \cdot 2^{- \frac{t}{180}} = 1 \cdot 40$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $40$ by $1$ .

$13 \cdot 2^{- \frac{t}{180}} = 40$

Step 5.4

Solve for $t$ .

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Step 5.4.1

Divide each term in $13 \cdot 2^{- \frac{t}{180}} = 40$ by $13$ and simplify.

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Step 5.4.1.1

Divide each term in $13 \cdot 2^{- \frac{t}{180}} = 40$ by $13$ .

$\frac{13 \cdot 2^{- \frac{t}{180}}}{13} = \frac{40}{13}$

Step 5.4.1.2

Simplify the left side.

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Step 5.4.1.2.1

Cancel the common factor of $13$ .

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Step 5.4.1.2.1.1

Cancel the common factor.

$\frac{13 \cdot 2^{- \frac{t}{180}}}{13} = \frac{40}{13}$

Step 5.4.1.2.1.2

Divide $2^{- \frac{t}{180}}$ by $1$ .

$2^{- \frac{t}{180}} = \frac{40}{13}$

Step 5.4.2

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

$\ln (2^{- \frac{t}{180}}) = \ln (\frac{40}{13})$

Step 5.4.3

Expand the left side.

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Step 5.4.3.1

Expand $\ln (2^{- \frac{t}{180}})$ by moving $- \frac{t}{180}$ outside the logarithm.

$- \frac{t}{180} \ln (2) = \ln (\frac{40}{13})$

Step 5.4.3.2

Combine $\ln (2)$ and $\frac{t}{180}$ .

$- \frac{\ln (2) t}{180} = \ln (\frac{40}{13})$

Step 5.4.4

Simplify the left side.

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Step 5.4.4.1

Reorder factors in $- \frac{\ln (2) t}{180}$ .

$- \frac{t \ln (2)}{180} = \ln (\frac{40}{13})$

Step 5.4.5

Multiply both sides of the equation by $- \frac{180}{\ln (2)}$ .

$- \frac{180}{\ln (2)} (- \frac{t \ln (2)}{180}) = - \frac{180}{\ln (2)} \ln (\frac{40}{13})$

Step 5.4.6

Simplify both sides of the equation.

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Step 5.4.6.1

Simplify the left side.

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Step 5.4.6.1.1

Simplify $- \frac{180}{\ln (2)} (- \frac{t \ln (2)}{180})$ .

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Step 5.4.6.1.1.1

Cancel the common factor of $180$ .

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Step 5.4.6.1.1.1.1

Move the leading negative in $- \frac{180}{\ln (2)}$ into the numerator.

$\frac{- 180}{\ln (2)} (- \frac{t \ln (2)}{180}) = - \frac{180}{\ln (2)} \ln (\frac{40}{13})$

Step 5.4.6.1.1.1.2

Move the leading negative in $- \frac{t \ln (2)}{180}$ into the numerator.

$\frac{- 180}{\ln (2)} \cdot \frac{- t \ln (2)}{180} = - \frac{180}{\ln (2)} \ln (\frac{40}{13})$

Step 5.4.6.1.1.1.3

Factor $180$ out of $- 180$ .

$\frac{180 (- 1)}{\ln (2)} \cdot \frac{- t \ln (2)}{180} = - \frac{180}{\ln (2)} \ln (\frac{40}{13})$

Step 5.4.6.1.1.1.4

Cancel the common factor.

$\frac{180 \cdot - 1}{\ln (2)} \cdot \frac{- t \ln (2)}{180} = - \frac{180}{\ln (2)} \ln (\frac{40}{13})$

Step 5.4.6.1.1.1.5

Rewrite the expression.

$\frac{- 1}{\ln (2)} (- t \ln (2)) = - \frac{180}{\ln (2)} \ln (\frac{40}{13})$

Step 5.4.6.1.1.2

Cancel the common factor of $\ln (2)$ .

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Step 5.4.6.1.1.2.1

Factor $\ln (2)$ out of $- t \ln (2)$ .

$\frac{- 1}{\ln (2)} (\ln (2) (- t)) = - \frac{180}{\ln (2)} \ln (\frac{40}{13})$

Step 5.4.6.1.1.2.2

Cancel the common factor.

$\frac{- 1}{\ln (2)} (\ln (2) (- t)) = - \frac{180}{\ln (2)} \ln (\frac{40}{13})$

Step 5.4.6.1.1.2.3

Rewrite the expression.

$- - t = - \frac{180}{\ln (2)} \ln (\frac{40}{13})$

Step 5.4.6.1.1.3

Multiply.

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Step 5.4.6.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 t = - \frac{180}{\ln (2)} \ln (\frac{40}{13})$

Step 5.4.6.1.1.3.2

Multiply $t$ by $1$ .

$t = - \frac{180}{\ln (2)} \ln (\frac{40}{13})$

Step 5.4.6.2

Simplify the right side.

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Step 5.4.6.2.1

Simplify $- \frac{180}{\ln (2)} \ln (\frac{40}{13})$ .

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Step 5.4.6.2.1.1

Combine $\ln (\frac{40}{13})$ and $\frac{180}{\ln (2)}$ .

$t = - \frac{\ln (\frac{40}{13}) \cdot 180}{\ln (2)}$

Step 5.4.6.2.1.2

Move $180$ to the left of $\ln (\frac{40}{13})$ .

$t = - \frac{180 \ln (\frac{40}{13})}{\ln (2)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = - \frac{180 \ln (\frac{40}{13})}{\ln (2)}$

Decimal Form:

$t = - 291.86790781 \dots$