Find the Inverse y=5^x-9

$y = 5^{x} - 9$

Step 1

Interchange the variables.

$x = 5^{y} - 9$

Step 2

Solve for $y$ .

Rewrite the equation as $5^{y} - 9 = x$ .

$5^{y} - 9 = x$

Add $9$ to both sides of the equation.

$5^{y} = x + 9$

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

$\ln (5^{y}) = \ln (x + 9)$

Expand $\ln (5^{y})$ by moving $y$ outside the logarithm.

$y \ln (5) = \ln (x + 9)$

Divide each term in $y \ln (5) = \ln (x + 9)$ by $\ln (5)$ and simplify.

Divide each term in $y \ln (5) = \ln (x + 9)$ by $\ln (5)$ .

$\frac{y \ln (5)}{\ln (5)} = \frac{\ln (x + 9)}{\ln (5)}$

Simplify the left side.

Cancel the common factor of $\ln (5)$ .

Cancel the common factor.

$\frac{y \ln (5)}{\ln (5)} = \frac{\ln (x + 9)}{\ln (5)}$

Divide $y$ by $1$ .

$y = \frac{\ln (x + 9)}{\ln (5)}$

Step 3

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

$f^{- 1} (x) = \frac{\ln (x + 9)}{\ln (5)}$

Step 4

Verify if $f^{- 1} (x) = \frac{\ln (x + 9)}{\ln (5)}$ is the inverse of $f (x) = 5^{x} - 9$ .

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

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

Set up the composite result function.

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

Evaluate $f^{- 1} (5^{x} - 9)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (5^{x} - 9) = \frac{\ln ((5^{x} - 9) + 9)}{\ln (5)}$

Combine the opposite terms in $5^{x} - 9 + 9$ .

Add $- 9$ and $9$ .

$f^{- 1} (5^{x} - 9) = \frac{\ln (5^{x} + 0)}{\ln (5)}$

Add $5^{x}$ and $0$ .

$f^{- 1} (5^{x} - 9) = \frac{\ln (5^{x})}{\ln (5)}$

Expand $\ln (5^{x})$ by moving $x$ outside the logarithm.

$f^{- 1} (5^{x} - 9) = \frac{x \ln (5)}{\ln (5)}$

Cancel the common factor of $\ln (5)$ .

Cancel the common factor.

$f^{- 1} (5^{x} - 9) = \frac{x \ln (5)}{\ln (5)}$

Divide $x$ by $1$ .

$f^{- 1} (5^{x} - 9) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{\ln (x + 9)}{\ln (5)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\ln (x + 9)}{\ln (5)}) = 5^{\frac{\ln (x + 9)}{\ln (5)}} - 9$

Simplify each term.

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

$f (\frac{\ln (x + 9)}{\ln (5)}) = 5^{\log_{5} (x + 9)} - 9$

Exponentiation and log are inverse functions.

$f (\frac{\ln (x + 9)}{\ln (5)}) = x + 9 - 9$

Combine the opposite terms in $x + 9 - 9$ .

Subtract $9$ from $9$ .

$f (\frac{\ln (x + 9)}{\ln (5)}) = x + 0$

Add $x$ and $0$ .

$f (\frac{\ln (x + 9)}{\ln (5)}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\ln (x + 9)}{\ln (5)}$ is the inverse of $f (x) = 5^{x} - 9$ .

$f^{- 1} (x) = \frac{\ln (x + 9)}{\ln (5)}$

Find the Inverse y=5^x-9

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