Finite Math Examples

$f (x) = 5 \log (3) x$

Step 1

Write $f (x) = 5 \log (3) x$ as an equation.

$y = 5 \log (3) x$

Step 2

Interchange the variables.

$x = 5 \log (3) y$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $5 \log (3) y = x$ .

$5 \log (3) y = x$

Step 3.2

Divide each term in $5 \log (3) y = x$ by $5 \log (3)$ and simplify.

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Step 3.2.1

Divide each term in $5 \log (3) y = x$ by $5 \log (3)$ .

$\frac{5 \log (3) y}{5 \log (3)} = \frac{x}{5 \log (3)}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 \log (3) y}{5 \log (3)} = \frac{x}{5 \log (3)}$

Step 3.2.2.1.2

Rewrite the expression.

$\frac{\log (3) y}{\log (3)} = \frac{x}{5 \log (3)}$

Step 3.2.2.2

Cancel the common factor of $\log (3)$ .

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Step 3.2.2.2.1

Cancel the common factor.

$\frac{\log (3) y}{\log (3)} = \frac{x}{5 \log (3)}$

Step 3.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{x}{5 \log (3)}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify $5 \log (3)$ by moving $5$ inside the logarithm.

$y = \frac{x}{\log (3^{5})}$

Step 3.2.3.2

Raise $3$ to the power of $5$ .

$y = \frac{x}{\log (243)}$

Step 4

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

$f^{- 1} (x) = \frac{x}{\log (243)}$

Step 5

Verify if $f^{- 1} (x) = \frac{x}{\log (243)}$ is the inverse of $f (x) = 5 \log (3) x$ .

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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 $f^{- 1} (5 \log (3) x)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (5 \log (3) x) = \frac{5 \log (3) x}{\log (243)}$

Step 5.2.3

Simplify $5 \log (3)$ by moving $5$ inside the logarithm.

$f^{- 1} (5 \log (3) x) = \frac{\log (3^{5}) x}{\log (243)}$

Step 5.2.4

Raise $3$ to the power of $5$ .

$f^{- 1} (5 \log (3) x) = \frac{\log (243) x}{\log (243)}$

Step 5.2.5

Cancel the common factor of $\log (243)$ .

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Step 5.2.5.1

Cancel the common factor.

$f^{- 1} (5 \log (3) x) = \frac{\log (243) x}{\log (243)}$

Step 5.2.5.2

Divide $x$ by $1$ .

$f^{- 1} (5 \log (3) x) = x$

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 $f (\frac{x}{\log (243)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{\log (243)}) = 5 \log (3) (\frac{x}{\log (243)})$

Step 5.3.3

Simplify $5 \log (3)$ by moving $5$ inside the logarithm.

$f (\frac{x}{\log (243)}) = \log (3^{5}) (\frac{x}{\log (243)})$

Step 5.3.4

Raise $3$ to the power of $5$ .

$f (\frac{x}{\log (243)}) = \log (243) (\frac{x}{\log (243)})$

Step 5.3.5

Cancel the common factor of $\log (243)$ .

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Step 5.3.5.1

Cancel the common factor.

$f (\frac{x}{\log (243)}) = \log (243) (\frac{x}{\log (243)})$

Step 5.3.5.2

Rewrite the expression.

$f (\frac{x}{\log (243)}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{\log (243)}$ is the inverse of $f (x) = 5 \log (3) x$ .

$f^{- 1} (x) = \frac{x}{\log (243)}$