Precalculus Examples

$f (x) = \ln (\ln (x))$

Step 1

Write $f (x) = \ln (\ln (x))$ as an equation.

$y = \ln (\ln (x))$

Step 2

Interchange the variables.

$x = \ln (\ln (y))$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\ln (\ln (y)) = x$ .

$\ln (\ln (y)) = x$

Step 3.2

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (\ln (y))} = e^{x}$

Step 3.3

Rewrite $\ln (\ln (y)) = x$ 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 $b^{y} = x$ .

$e^{x} = \ln (y)$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Rewrite the equation as $\ln (y) = e^{x}$ .

$\ln (y) = e^{x}$

Step 3.4.2

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (y)} = e^{e^{x}}$

Step 3.4.3

Rewrite $\ln (y) = e^{x}$ 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 $b^{y} = x$ .

$e^{e^{x}} = y$

Step 3.4.4

Rewrite the equation as $y = e^{e^{x}}$ .

$y = e^{e^{x}}$

Step 4

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

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

Step 5

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

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

$f^{- 1} (\ln (\ln (x))) = e^{e^{\ln (\ln (x))}}$

Step 5.2.3

Exponentiation and log are inverse functions.

$f^{- 1} (\ln (\ln (x))) = e^{\ln (x)}$

Step 5.2.4

Exponentiation and log are inverse functions.

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

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

$f (e^{e^{x}}) = \ln (\ln (e^{e^{x}}))$

Step 5.3.3

Use logarithm rules to move $e^{x}$ out of the exponent.

$f (e^{e^{x}}) = \ln (e^{x} \ln (e))$

Step 5.3.4

The natural logarithm of $e$ is $1$ .

$f (e^{e^{x}}) = \ln (e^{x} \cdot 1)$

Step 5.3.5

Multiply $e^{x}$ by $1$ .

$f (e^{e^{x}}) = \ln (e^{x})$

Step 5.3.6

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

$f (e^{e^{x}}) = x \ln (e)$

Step 5.3.7

The natural logarithm of $e$ is $1$ .

$f (e^{e^{x}}) = x \cdot 1$

Step 5.3.8

Multiply $x$ by $1$ .

$f (e^{e^{x}}) = x$

Step 5.4

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

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