Find the Inverse f(x)=9^(6(x+2))

$f (x) = 9^{6 (x + 2)}$

Write $f (x) = 9^{6 (x + 2)}$ as an equation.

$y = 9^{6 (x + 2)}$

Interchange the variables.

$x = 9^{6 (y + 2)}$

Solve for $y$ .

Rewrite the equation as $9^{6 (y + 2)} = x$ .

$9^{6 (y + 2)} = x$

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

$\ln (9^{6 (y + 2)}) = \ln (x)$

Expand $\ln (9^{6 (y + 2)})$ by moving $6 (y + 2)$ outside the logarithm.

$6 (y + 2) \ln (9) = \ln (x)$

Simplify the left side.

Simplify $6 (y + 2) \ln (9)$ .

Apply the distributive property.

$(6 y + 6 \cdot 2) \ln (9) = \ln (x)$

Multiply $6$ by $2$ .

$(6 y + 12) \ln (9) = \ln (x)$

Apply the distributive property.

$6 y \ln (9) + 12 \ln (9) = \ln (x)$

Move all the terms containing a logarithm to the left side of the equation.

$6 y \ln (9) + 12 \ln (9) - \ln (x) = 0$

Move all terms not containing $y$ to the right side of the equation.

Subtract $12 \ln (9)$ from both sides of the equation.

$6 y \ln (9) - \ln (x) = - 12 \ln (9)$

Add $\ln (x)$ to both sides of the equation.

$6 y \ln (9) = - 12 \ln (9) + \ln (x)$

Divide each term in $6 y \ln (9) = - 12 \ln (9) + \ln (x)$ by $6 \ln (9)$ and simplify.

Divide each term in $6 y \ln (9) = - 12 \ln (9) + \ln (x)$ by $6 \ln (9)$ .

$\frac{6 y \ln (9)}{6 \ln (9)} = \frac{- 12 \ln (9)}{6 \ln (9)} + \frac{\ln (x)}{6 \ln (9)}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y \ln (9)}{6 \ln (9)} = \frac{- 12 \ln (9)}{6 \ln (9)} + \frac{\ln (x)}{6 \ln (9)}$

Rewrite the expression.

$\frac{y \ln (9)}{\ln (9)} = \frac{- 12 \ln (9)}{6 \ln (9)} + \frac{\ln (x)}{6 \ln (9)}$

Cancel the common factor of $\ln (9)$ .

Cancel the common factor.

$\frac{y \ln (9)}{\ln (9)} = \frac{- 12 \ln (9)}{6 \ln (9)} + \frac{\ln (x)}{6 \ln (9)}$

Divide $y$ by $1$ .

$y = \frac{- 12 \ln (9)}{6 \ln (9)} + \frac{\ln (x)}{6 \ln (9)}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 12$ and $6$ .

Factor $6$ out of $- 12 \ln (9)$ .

$y = \frac{6 (- 2 \ln (9))}{6 \ln (9)} + \frac{\ln (x)}{6 \ln (9)}$

Cancel the common factors.

Factor $6$ out of $6 \ln (9)$ .

$y = \frac{6 (- 2 \ln (9))}{6 \ln (9)} + \frac{\ln (x)}{6 \ln (9)}$

Cancel the common factor.

$y = \frac{6 (- 2 \ln (9))}{6 \ln (9)} + \frac{\ln (x)}{6 \ln (9)}$

Rewrite the expression.

$y = \frac{- 2 \ln (9)}{\ln (9)} + \frac{\ln (x)}{6 \ln (9)}$

Cancel the common factor of $\ln (9)$ .

Cancel the common factor.

$y = \frac{- 2 \ln (9)}{\ln (9)} + \frac{\ln (x)}{6 \ln (9)}$

Divide $- 2$ by $1$ .

$y = - 2 + \frac{\ln (x)}{6 \ln (9)}$

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

$f^{- 1} (x) = - 2 + \frac{\ln (x)}{6 \ln (9)}$

Verify if $f^{- 1} (x) = - 2 + \frac{\ln (x)}{6 \ln (9)}$ is the inverse of $f (x) = 9^{6 (x + 2)}$ .

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 $f^{- 1} \cdot 9^{6 (x + 2)}$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} \cdot 9^{6 (x + 2)} = - 2 + \frac{\ln (9^{6 (x + 2)})}{6 \ln (9)}$

Simplify each term.

Expand $\ln (9^{6 (x + 2)})$ by moving $6 (x + 2)$ outside the logarithm.

$f^{- 1} \cdot 9^{6 (x + 2)} = - 2 + \frac{6 (x + 2) \ln (9)}{6 \ln (9)}$

Cancel the common factor of $6$ .

Cancel the common factor.

$f^{- 1} \cdot 9^{6 (x + 2)} = - 2 + \frac{6 (x + 2) \ln (9)}{6 \ln (9)}$

Rewrite the expression.

$f^{- 1} \cdot 9^{6 (x + 2)} = - 2 + \frac{(x + 2) \ln (9)}{\ln (9)}$

Cancel the common factor of $\ln (9)$ .

Cancel the common factor.

$f^{- 1} \cdot 9^{6 (x + 2)} = - 2 + \frac{(x + 2) \ln (9)}{\ln (9)}$

Divide $x + 2$ by $1$ .

$f^{- 1} \cdot 9^{6 (x + 2)} = - 2 + x + 2$

Combine the opposite terms in $- 2 + x + 2$ .

Add $- 2$ and $2$ .

$f^{- 1} \cdot 9^{6 (x + 2)} = x + 0$

Add $x$ and $0$ .

$f^{- 1} \cdot 9^{6 (x + 2)} = x$

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

Set up the composite result function.

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

Evaluate $f (- 2 + \frac{\ln (x)}{6 \ln (9)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- 2 + \frac{\ln (x)}{6 \ln (9)}) = 9^{6 ((- 2 + \frac{\ln (x)}{6 \ln (9)}) + 2)}$

Combine the opposite terms in $(- 2 + \frac{\ln (x)}{6 \ln (9)}) + 2$ .

Add $- 2$ and $2$ .

$f (- 2 + \frac{\ln (x)}{6 \ln (9)}) = 9^{6 (0 + \frac{\ln (x)}{6 \ln (9)})}$

Add $0$ and $\frac{\ln (x)}{6 \ln (9)}$ .

$f (- 2 + \frac{\ln (x)}{6 \ln (9)}) = 9^{6 (\frac{\ln (x)}{6 \ln (9)})}$

Simplify $6 \ln (9)$ by moving $6$ inside the logarithm.

$f (- 2 + \frac{\ln (x)}{6 \ln (9)}) = 9^{6 (\frac{\ln (x)}{\ln (9^{6})})}$

Raise $9$ to the power of $6$ .

$f (- 2 + \frac{\ln (x)}{6 \ln (9)}) = 9^{6 (\frac{\ln (x)}{\ln (531441)})}$

Multiply $6 \frac{\ln (x)}{\ln (531441)}$ .

Combine $6$ and $\frac{\ln (x)}{\ln (531441)}$ .

$f (- 2 + \frac{\ln (x)}{6 \ln (9)}) = 9^{\frac{6 \ln (x)}{\ln (531441)}}$

Simplify $6 \ln (x)$ by moving $6$ inside the logarithm.

$f (- 2 + \frac{\ln (x)}{6 \ln (9)}) = 9^{\frac{\ln (x^{6})}{\ln (531441)}}$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - 2 + \frac{\ln (x)}{6 \ln (9)}$ is the inverse of $f (x) = 9^{6 (x + 2)}$ .

$f^{- 1} (x) = - 2 + \frac{\ln (x)}{6 \ln (9)}$

Find the Inverse f(x)=9^(6(x+2))

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