Algebra Examples

$f (x) = 50000 {(0.8)}^{x}$

Step 1

Write $f (x) = 50000 {(0.8)}^{x}$ as an equation.

$y = 50000 {(0.8)}^{x}$

Step 2

Interchange the variables.

$x = 50000 {(0.8)}^{y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $50000 \cdot {0.8}^{y} = x$ .

$50000 \cdot {0.8}^{y} = x$

Step 3.2

Divide each term in $50000 \cdot {0.8}^{y} = x$ by $50000$ and simplify.

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Step 3.2.1

Divide each term in $50000 \cdot {0.8}^{y} = x$ by $50000$ .

$\frac{50000 \cdot {0.8}^{y}}{50000} = \frac{x}{50000}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $50000$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{50000 \cdot {0.8}^{y}}{50000} = \frac{x}{50000}$

Step 3.2.2.1.2

Divide ${0.8}^{y}$ by $1$ .

${0.8}^{y} = \frac{x}{50000}$

Step 3.3

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

$\ln ({0.8}^{y}) = \ln (\frac{x}{50000})$

Step 3.4

Expand $\ln ({0.8}^{y})$ by moving $y$ outside the logarithm.

$y \ln (0.8) = \ln (\frac{x}{50000})$

Step 3.5

Divide each term in $y \ln (0.8) = \ln (\frac{x}{50000})$ by $\ln (0.8)$ and simplify.

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Step 3.5.1

Divide each term in $y \ln (0.8) = \ln (\frac{x}{50000})$ by $\ln (0.8)$ .

$\frac{y \ln (0.8)}{\ln (0.8)} = \frac{\ln (\frac{x}{50000})}{\ln (0.8)}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $\ln (0.8)$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{y \ln (0.8)}{\ln (0.8)} = \frac{\ln (\frac{x}{50000})}{\ln (0.8)}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (\frac{x}{50000})}{\ln (0.8)}$

Step 4

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

$f^{- 1} (x) = \frac{\ln (\frac{x}{50000})}{\ln (0.8)}$

Step 5

Verify if $f^{- 1} (x) = \frac{\ln (\frac{x}{50000})}{\ln (0.8)}$ is the inverse of $f (x) = 50000 {(0.8)}^{x}$ .

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Step 5.1

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

Step 5.2

Evaluate $f^{- 1} (f (x))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate $f^{- 1} (50000 {(0.8)}^{x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (50000 {(0.8)}^{x}) = \frac{\ln (\frac{50000 {(0.8)}^{x}}{50000})}{\ln (0.8)}$

Step 5.2.3

Cancel the common factor of $50000$ .

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Step 5.2.3.1

Cancel the common factor.

$f^{- 1} (50000 {(0.8)}^{x}) = \frac{\ln (\frac{50000 {(0.8)}^{x}}{50000})}{\ln (0.8)}$

Step 5.2.3.2

Divide ${(0.8)}^{x}$ by $1$ .

$f^{- 1} (50000 {(0.8)}^{x}) = \frac{\ln ({(0.8)}^{x})}{\ln (0.8)}$

Step 5.2.4

Expand $\ln ({(0.8)}^{x})$ by moving $x$ outside the logarithm.

$f^{- 1} (50000 {(0.8)}^{x}) = \frac{x \ln (0.8)}{\ln (0.8)}$

Step 5.2.5

Cancel the common factor of $\ln (0.8)$ .

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Step 5.2.5.1

Cancel the common factor.

$f^{- 1} (50000 {(0.8)}^{x}) = \frac{x \ln (0.8)}{\ln (0.8)}$

Step 5.2.5.2

Divide $x$ by $1$ .

$f^{- 1} (50000 {(0.8)}^{x}) = x$

Step 5.3

Evaluate $f (f^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate $f (\frac{\ln (\frac{x}{50000})}{\ln (0.8)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\ln (\frac{x}{50000})}{\ln (0.8)}) = 50000 {(0.8)}^{\frac{\ln (\frac{x}{50000})}{\ln (0.8)}}$

Step 5.3.3

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

$f (\frac{\ln (\frac{x}{50000})}{\ln (0.8)}) = 50000 \cdot {0.8}^{\log_{0.8} (\frac{x}{50000})}$

Step 5.3.4

Exponentiation and log are inverse functions.

$f (\frac{\ln (\frac{x}{50000})}{\ln (0.8)}) = 50000 \cdot \frac{x}{50000}$

Step 5.3.5

Cancel the common factor of $50000$ .

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Step 5.3.5.1

Cancel the common factor.

$f (\frac{\ln (\frac{x}{50000})}{\ln (0.8)}) = 50000 \cdot \frac{x}{50000}$

Step 5.3.5.2

Rewrite the expression.

$f (\frac{\ln (\frac{x}{50000})}{\ln (0.8)}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\ln (\frac{x}{50000})}{\ln (0.8)}$ is the inverse of $f (x) = 50000 {(0.8)}^{x}$ .

$f^{- 1} (x) = \frac{\ln (\frac{x}{50000})}{\ln (0.8)}$