Algebra Examples

$y = \log (x + 4)$

Interchange the variables.

$x = \log (y + 4)$

Solve for $y$ .

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Rewrite the equation as $\log (y + 4) = x$ .

$\log (y + 4) = x$

Rewrite $\log (y + 4) = x$ 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 $b^{y} = x$ .

$10^{x} = y + 4$

Solve for $y$ .

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Rewrite the equation as $y + 4 = 10^{x}$ .

$y + 4 = 10^{x}$

Subtract $4$ from both sides of the equation.

$y = 10^{x} - 4$

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

$f^{- 1} (x) = 10^{x} - 4$

Verify if $f^{- 1} (x) = 10^{x} - 4$ is the inverse of $f (x) = \log (x + 4)$ .

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

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

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Set up the composite result function.

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

Evaluate $f^{- 1} (\log (x + 4))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\log (x + 4)) = 10^{\log (x + 4)} - 4$

Exponentiation and log are inverse functions.

$f^{- 1} (\log (x + 4)) = x + 4 - 4$

Combine the opposite terms in $x + 4 - 4$ .

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Subtract $4$ from $4$ .

$f^{- 1} (\log (x + 4)) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\log (x + 4)) = x$

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

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Set up the composite result function.

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

Evaluate $f (10^{x} - 4)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (10^{x} - 4) = \log ((10^{x} - 4) + 4)$

Combine the opposite terms in $(10^{x} - 4) + 4$ .

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Add $- 4$ and $4$ .

$f (10^{x} - 4) = \log (10^{x} + 0)$

Add $10^{x}$ and $0$ .

$f (10^{x} - 4) = \log (10^{x})$

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

$f (10^{x} - 4) = x \log (10)$

Logarithm base $10$ of $10$ is $1$ .

$f (10^{x} - 4) = x \cdot 1$

Multiply $x$ by $1$ .

$f (10^{x} - 4) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 10^{x} - 4$ is the inverse of $f (x) = \log (x + 4)$ .

$f^{- 1} (x) = 10^{x} - 4$