Finite Math Examples

$f (x) = p \log (e)$

Step 1

Write $f (x) = p \log (e)$ as an equation.

$y = p \log (e)$

Step 2

Interchange the variables.

$p = y \log (e)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $y \log (e) = p$ .

$y \log (e) = p$

Step 3.2

Divide each term in $y \log (e) = p$ by $\log (e)$ and simplify.

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Step 3.2.1

Divide each term in $y \log (e) = p$ by $\log (e)$ .

$\frac{y \log (e)}{\log (e)} = \frac{p}{\log (e)}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $\log (e)$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{y \log (e)}{\log (e)} = \frac{p}{\log (e)}$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{p}{\log (e)}$

Step 4

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

$f^{- 1} (p) = \frac{p}{\log (e)}$

Step 5

Verify if $f^{- 1} (p) = \frac{p}{\log (e)}$ is the inverse of $f (p) = p \log (e)$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (p \log (e))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (p \log (e)) = \frac{p \log (e)}{\log (e)}$

Step 5.2.3

Cancel the common factor of $\log (e)$ .

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Step 5.2.3.1

Cancel the common factor.

$f^{- 1} (p \log (e)) = \frac{p \log (e)}{\log (e)}$

Step 5.2.3.2

Divide $p$ by $1$ .

$f^{- 1} (p \log (e)) = p$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{p}{\log (e)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{p}{\log (e)}) = (\frac{p}{\log (e)}) \log (e)$

Step 5.3.3

Cancel the common factor of $\log (e)$ .

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Step 5.3.3.1

Cancel the common factor.

$f (\frac{p}{\log (e)}) = \frac{p}{\log (e)} \cdot \log (e)$

Step 5.3.3.2

Rewrite the expression.

$f (\frac{p}{\log (e)}) = p$

Step 5.4

Since $f^{- 1} (f (p)) = p$ and $f (f^{- 1} (p)) = p$ , then $f^{- 1} (p) = \frac{p}{\log (e)}$ is the inverse of $f (p) = p \log (e)$ .

$f^{- 1} (p) = \frac{p}{\log (e)}$