Algebra Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Simplify the left side.

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Step 3.4.1

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

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Step 3.4.1.1

Apply the distributive property.

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

Step 3.4.1.2

Multiply $6$ by $2$ .

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

Step 3.4.1.3

Apply the distributive property.

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

Step 3.5

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

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

Simplify the left side.

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Step 3.7.2.1

Cancel the common factor of $6$ .

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Step 3.7.2.1.1

Cancel the common factor.

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

Step 3.7.2.1.2

Rewrite the expression.

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

Step 3.7.2.2

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

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Step 3.7.2.2.1

Cancel the common factor.

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

Step 3.7.2.2.2

Divide $y$ by $1$ .

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

Step 3.7.3

Simplify the right side.

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Step 3.7.3.1

Simplify each term.

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Step 3.7.3.1.1

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

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Step 3.7.3.1.1.1

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

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

Step 3.7.3.1.1.2

Cancel the common factors.

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Step 3.7.3.1.1.2.1

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

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

Step 3.7.3.1.1.2.2

Cancel the common factor.

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

Step 3.7.3.1.1.2.3

Rewrite the expression.

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

Step 3.7.3.1.2

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

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Step 3.7.3.1.2.1

Cancel the common factor.

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

Step 3.7.3.1.2.2

Divide $- 2$ by $1$ .

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

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

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

Step 5.2.3.2

Cancel the common factor of $6$ .

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Step 5.2.3.2.1

Cancel the common factor.

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

Step 5.2.3.2.2

Rewrite the expression.

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

Step 5.2.3.3

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

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Step 5.2.3.3.1

Cancel the common factor.

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

Step 5.2.3.3.2

Divide $x + 2$ by $1$ .

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

Step 5.2.4

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

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Step 5.2.4.1

Add $- 2$ and $2$ .

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

Step 5.2.4.2

Add $x$ and $0$ .

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

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

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

Step 5.3.3

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

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Step 5.3.3.1

Add $- 2$ and $2$ .

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

Step 5.3.3.2

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

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

Step 5.3.4

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

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

Step 5.3.5

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

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

Step 5.3.6

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

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Step 5.3.6.1

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

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

Step 5.3.6.2

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

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

Step 5.4

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

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