Trigonometry Examples

$10000 = 20000 {(1.3)}^{- 0.234 \cdot t}$

Step 1

Rewrite the equation as $20000 \cdot {1.3}^{- 0.234 \cdot t} = 10000$ .

$20000 \cdot {1.3}^{- 0.234 \cdot t} = 10000$

Step 2

Divide each term in $20000 \cdot {1.3}^{- 0.234 \cdot t} = 10000$ by $20000$ and simplify.

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Step 2.1

Divide each term in $20000 \cdot {1.3}^{- 0.234 \cdot t} = 10000$ by $20000$ .

$\frac{20000 \cdot {1.3}^{- 0.234 \cdot t}}{20000} = \frac{10000}{20000}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $20000$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{20000 \cdot {1.3}^{- 0.234 \cdot t}}{20000} = \frac{10000}{20000}$

Step 2.2.1.2

Divide ${1.3}^{- 0.234 \cdot t}$ by $1$ .

${1.3}^{- 0.234 \cdot t} = \frac{10000}{20000}$

${1.3}^{- 0.234 t} = \frac{10000}{20000}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $10000$ and $20000$ .

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Step 2.3.1.1

Factor $10000$ out of $10000$ .

${1.3}^{- 0.234 t} = \frac{10000 (1)}{20000}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $10000$ out of $20000$ .

${1.3}^{- 0.234 t} = \frac{10000 \cdot 1}{10000 \cdot 2}$

Step 2.3.1.2.2

Cancel the common factor.

${1.3}^{- 0.234 t} = \frac{10000 \cdot 1}{10000 \cdot 2}$

Step 2.3.1.2.3

Rewrite the expression.

${1.3}^{- 0.234 t} = \frac{1}{2}$

Step 3

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

$\ln ({1.3}^{- 0.234 t}) = \ln (\frac{1}{2})$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln ({1.3}^{- 0.234 t})$ by moving $- 0.234 t$ outside the logarithm.

$- 0.234 t \ln (1.3) = \ln (\frac{1}{2})$

Step 4.2

Multiply $\ln (1.3)$ by $- 0.234$ .

$- 0.06139323 t = \ln (\frac{1}{2})$

Step 5

Divide each term in $- 0.06139323 t = \ln (\frac{1}{2})$ by $- 0.06139323$ and simplify.

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Step 5.1

Divide each term in $- 0.06139323 t = \ln (\frac{1}{2})$ by $- 0.06139323$ .

$\frac{- 0.06139323 t}{- 0.06139323} = \frac{\ln (\frac{1}{2})}{- 0.06139323}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 0.06139323$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 0.06139323 t}{- 0.06139323} = \frac{\ln (\frac{1}{2})}{- 0.06139323}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{1}{2})}{- 0.06139323}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (\frac{1}{2})}{0.06139323}$

Step 5.3.2

Replace $e$ with an approximation.

$t = - \frac{\log_{2.71828182} (\frac{1}{2})}{0.06139323}$

Step 5.3.3

Divide $1$ by $2$ .

$t = - \frac{\log_{2.71828182} (0.5)}{0.06139323}$

Step 5.3.4

Log base $2.71828182$ of $0.5$ is approximately $- 0.69314718$ .

$t = - \frac{- 0.69314718}{0.06139323}$

Step 5.3.5

Divide $- 0.69314718$ by $0.06139323$ .

$t = - - 11.29028545$

Step 5.3.6

Multiply $- 1$ by $- 11.29028545$ .

$t = 11.29028545$