Finite Math Examples

$P (t) = 20000 {(1.02)}^{t}$

Step 1

Write $P (t) = 20000 {(1.02)}^{t}$ as an equation.

$y = 20000 {(1.02)}^{t}$

Step 2

Interchange the variables.

$t = 20000 {(1.02)}^{y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $20000 \cdot {1.02}^{y} = t$ .

$20000 \cdot {1.02}^{y} = t$

Step 3.2

Divide each term in $20000 \cdot {1.02}^{y} = t$ by $20000$ and simplify.

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Step 3.2.1

Divide each term in $20000 \cdot {1.02}^{y} = t$ by $20000$ .

$\frac{20000 \cdot {1.02}^{y}}{20000} = \frac{t}{20000}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $20000$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{20000 \cdot {1.02}^{y}}{20000} = \frac{t}{20000}$

Step 3.2.2.1.2

Divide ${1.02}^{y}$ by $1$ .

${1.02}^{y} = \frac{t}{20000}$

Step 3.3

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

$\ln ({1.02}^{y}) = \ln (\frac{t}{20000})$

Step 3.4

Expand $\ln ({1.02}^{y})$ by moving $y$ outside the logarithm.

$y \ln (1.02) = \ln (\frac{t}{20000})$

Step 3.5

Divide each term in $y \ln (1.02) = \ln (\frac{t}{20000})$ by $\ln (1.02)$ and simplify.

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Step 3.5.1

Divide each term in $y \ln (1.02) = \ln (\frac{t}{20000})$ by $\ln (1.02)$ .

$\frac{y \ln (1.02)}{\ln (1.02)} = \frac{\ln (\frac{t}{20000})}{\ln (1.02)}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $\ln (1.02)$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{y \ln (1.02)}{\ln (1.02)} = \frac{\ln (\frac{t}{20000})}{\ln (1.02)}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (\frac{t}{20000})}{\ln (1.02)}$

Step 4

Replace $y$ with $P^{- 1} (t)$ to show the final answer.

$P^{- 1} (t) = \frac{\ln (\frac{t}{20000})}{\ln (1.02)}$

Step 5

Verify if $P^{- 1} (t) = \frac{\ln (\frac{t}{20000})}{\ln (1.02)}$ is the inverse of $P (t) = 20000 {(1.02)}^{t}$ .

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Step 5.1

To verify the inverse, check if $P^{- 1} (P (t)) = t$ and $P (P^{- 1} (t)) = t$ .

Step 5.2

Evaluate $P^{- 1} (P (t))$ .

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Step 5.2.1

Set up the composite result function.

$P^{- 1} (P (t))$

Step 5.2.2

Evaluate $P^{- 1} (20000 {(1.02)}^{t})$ by substituting in the value of $P$ into $P^{- 1}$ .

$P^{- 1} (20000 {(1.02)}^{t}) = \frac{\ln (\frac{20000 {(1.02)}^{t}}{20000})}{\ln (1.02)}$

Step 5.2.3

Cancel the common factor of $20000$ .

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Step 5.2.3.1

Cancel the common factor.

$P^{- 1} (20000 {(1.02)}^{t}) = \frac{\ln (\frac{20000 {(1.02)}^{t}}{20000})}{\ln (1.02)}$

Step 5.2.3.2

Divide ${(1.02)}^{t}$ by $1$ .

$P^{- 1} (20000 {(1.02)}^{t}) = \frac{\ln ({(1.02)}^{t})}{\ln (1.02)}$

Step 5.2.4

Expand $\ln ({(1.02)}^{t})$ by moving $t$ outside the logarithm.

$P^{- 1} (20000 {(1.02)}^{t}) = \frac{t \ln (1.02)}{\ln (1.02)}$

Step 5.2.5

Cancel the common factor of $\ln (1.02)$ .

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Step 5.2.5.1

Cancel the common factor.

$P^{- 1} (20000 {(1.02)}^{t}) = \frac{t \ln (1.02)}{\ln (1.02)}$

Step 5.2.5.2

Divide $t$ by $1$ .

$P^{- 1} (20000 {(1.02)}^{t}) = t$

Step 5.3

Evaluate $P (P^{- 1} (t))$ .

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Step 5.3.1

Set up the composite result function.

$P (P^{- 1} (t))$

Step 5.3.2

Evaluate $P (\frac{\ln (\frac{t}{20000})}{\ln (1.02)})$ by substituting in the value of $P^{- 1}$ into $P$ .

$P (\frac{\ln (\frac{t}{20000})}{\ln (1.02)}) = 20000 {(1.02)}^{\frac{\ln (\frac{t}{20000})}{\ln (1.02)}}$

Step 5.3.3

Use the change of base rule $\frac{\log_{b} (x)}{\log_{b} (a)} = \log_{a} (x)$ .

$P (\frac{\ln (\frac{t}{20000})}{\ln (1.02)}) = 20000 \cdot {1.02}^{\log_{1.02} (\frac{t}{20000})}$

Step 5.3.4

Exponentiation and log are inverse functions.

$P (\frac{\ln (\frac{t}{20000})}{\ln (1.02)}) = 20000 \cdot \frac{t}{20000}$

Step 5.3.5

Cancel the common factor of $20000$ .

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Step 5.3.5.1

Cancel the common factor.

$P (\frac{\ln (\frac{t}{20000})}{\ln (1.02)}) = 20000 \cdot \frac{t}{20000}$

Step 5.3.5.2

Rewrite the expression.

$P (\frac{\ln (\frac{t}{20000})}{\ln (1.02)}) = t$

Step 5.4

Since $P^{- 1} (P (t)) = t$ and $P (P^{- 1} (t)) = t$ , then $P^{- 1} (t) = \frac{\ln (\frac{t}{20000})}{\ln (1.02)}$ is the inverse of $P (t) = 20000 {(1.02)}^{t}$ .

$P^{- 1} (t) = \frac{\ln (\frac{t}{20000})}{\ln (1.02)}$