Finite Math Examples

$f (x) = \log_{\frac{1}{5}} (x)$

Step 1

Write $f (x) = \log_{\frac{1}{5}} (x)$ as an equation.

$y = \log_{\frac{1}{5}} (x)$

Step 2

Interchange the variables.

$x = \log_{\frac{1}{5}} (y)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\log_{\frac{1}{5}} (y) = x$ .

$\log_{\frac{1}{5}} (y) = x$

Step 3.2

Rewrite $\log_{\frac{1}{5}} (y) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

${(\frac{1}{5})}^{x} = y$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $y = {(\frac{1}{5})}^{x}$ .

$y = {(\frac{1}{5})}^{x}$

Step 3.3.2

Simplify ${(\frac{1}{5})}^{x}$ .

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Step 3.3.2.1

Apply the product rule to $\frac{1}{5}$ .

$y = \frac{1^{x}}{5^{x}}$

Step 3.3.2.2

One to any power is one.

$y = \frac{1}{5^{x}}$

Step 4

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

$f^{- 1} (x) = \frac{1}{5^{x}}$

Step 5

Verify if $f^{- 1} (x) = \frac{1}{5^{x}}$ is the inverse of $f (x) = \log_{\frac{1}{5}} (x)$ .

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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 $f^{- 1} (\log_{\frac{1}{5}} (x))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\log_{\frac{1}{5}} (x)) = \frac{1}{5^{\log_{\frac{1}{5}} (x)}}$

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 $f (\frac{1}{5^{x}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{1}{5^{x}}) = \log_{\frac{1}{5}} (\frac{1}{5^{x}})$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{1}{5^{x}}$ is the inverse of $f (x) = \log_{\frac{1}{5}} (x)$ .

$f^{- 1} (x) = \frac{1}{5^{x}}$