Basic Math Examples

$32 = (70) \cdot {2.718282}^{- 0.122 t}$

Step 1

Rewrite the equation as $70 \cdot {2.718282}^{- 0.122 t} = 32$ .

$70 \cdot {2.718282}^{- 0.122 t} = 32$

Step 2

Divide each term in $70 \cdot {2.718282}^{- 0.122 t} = 32$ by $70$ and simplify.

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Step 2.1

Divide each term in $70 \cdot {2.718282}^{- 0.122 t} = 32$ by $70$ .

$\frac{70 \cdot {2.718282}^{- 0.122 t}}{70} = \frac{32}{70}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $70$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{70 \cdot {2.718282}^{- 0.122 t}}{70} = \frac{32}{70}$

Step 2.2.1.2

Divide ${2.718282}^{- 0.122 t}$ by $1$ .

${2.718282}^{- 0.122 t} = \frac{32}{70}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $32$ and $70$ .

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Step 2.3.1.1

Factor $2$ out of $32$ .

${2.718282}^{- 0.122 t} = \frac{2 (16)}{70}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $70$ .

${2.718282}^{- 0.122 t} = \frac{2 \cdot 16}{2 \cdot 35}$

Step 2.3.1.2.2

Cancel the common factor.

${2.718282}^{- 0.122 t} = \frac{2 \cdot 16}{2 \cdot 35}$

Step 2.3.1.2.3

Rewrite the expression.

${2.718282}^{- 0.122 t} = \frac{16}{35}$

Step 3

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

$\ln ({2.718282}^{- 0.122 t}) = \ln (\frac{16}{35})$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln ({2.718282}^{- 0.122 t})$ by moving $- 0.122 t$ outside the logarithm.

$- 0.122 t \ln (2.718282) = \ln (\frac{16}{35})$

Step 4.2

Multiply $\ln (2.718282)$ by $- 0.122$ .

$- 0.122 t = \ln (\frac{16}{35})$

Step 5

Divide each term in $- 0.122 t = \ln (\frac{16}{35})$ by $- 0.122$ and simplify.

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Step 5.1

Divide each term in $- 0.122 t = \ln (\frac{16}{35})$ by $- 0.122$ .

$\frac{- 0.122 t}{- 0.122} = \frac{\ln (\frac{16}{35})}{- 0.122}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 0.122$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 0.122 t}{- 0.122} = \frac{\ln (\frac{16}{35})}{- 0.122}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{16}{35})}{- 0.122}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (\frac{16}{35})}{0.122}$

Step 5.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (\frac{16}{35})}{0.122}$

Step 5.3.3

Divide $16$ by $35$ .

$t = - \frac{\log_{2.71828182} (0.4\overline{571428})}{0.122}$

Step 5.3.4

Log base $2.71828182$ of $0.4\overline{571428}$ is approximately $- 0.78275933$ .

$t = - \frac{- 0.78275933}{0.122}$

Step 5.3.5

Divide $- 0.78275933$ by $0.122$ .

$t = - - 6.41605975$

Step 5.3.6

Multiply $- 1$ by $- 6.41605975$ .

$t = 6.41605975$