Find the Inverse y = log base 6 of x+2

$y = \log_{6} (x + 2)$

Step 1

Interchange the variables.

$x = \log_{6} (y + 2)$

Step 2

Solve for $y$ .

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

$\log_{6} (y + 2) = x$

Rewrite $\log_{6} (y + 2) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$6^{x} = y + 2$

Solve for $y$ .

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

$y + 2 = 6^{x}$

Subtract $2$ from both sides of the equation.

$y = 6^{x} - 2$

Step 3

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

$f^{- 1} (x) = 6^{x} - 2$

Step 4

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

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

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

Exponentiation and log are inverse functions.

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

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

Subtract $2$ from $2$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

$f (6^{x} - 2) = \log_{6} ((6^{x} - 2) + 2)$

Combine the opposite terms in $(6^{x} - 2) + 2$ .

Add $- 2$ and $2$ .

$f (6^{x} - 2) = \log_{6} (6^{x} + 0)$

Add $6^{x}$ and $0$ .

$f (6^{x} - 2) = \log_{6} (6^{x})$

Use logarithm rules to move $x$ out of the exponent.

$f (6^{x} - 2) = x \log_{6} (6)$

Logarithm base $6$ of $6$ is $1$ .

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

Multiply $x$ by $1$ .

$f (6^{x} - 2) = x$

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

$f^{- 1} (x) = 6^{x} - 2$

Find the Inverse y = log base 6 of x+2

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