Problem 84 - Algebra and Trigonometry

Question

A function \(f(x)\) is given. (a) Find the domain of the function \(f .\) (b) Find the inverse function of \(f .\) $$ f(x)=\ln (\ln (\ln x)) $$

Step-by-Step Solution

Answer

The domain of \( f(x) \) is \( x > e \). The inverse function is \( f^{-1}(x) = e^{e^{e^x}} \).

Step 1: Determine the Domain of the Innermost Function

The innermost function is \( \ln(x) \). The natural logarithm function, \( \ln \), is defined only for positive values. Therefore, \( x > 0 \).

Step 2: Determine the Domain of the Second Function

The second function we consider is \( \ln(\ln(x)) \). For the function \( \ln(y) \) (with \( y = \ln(x) \)) to be defined, \( \ln(x) > 0 \), which implies that \( x > e^0 = 1 \).

Step 3: Determine the Domain of the Outermost Function

The outermost function is \( \ln(\ln(\ln x)) \). To ensure this is defined, \( \ln(\ln(x)) > 0 \) must hold. Therefore, \( \ln(x) > e^0 = 1 \), which results in \( x > e \).

Step 4: Write the Full Domain of \( f(x) \)

By solving through the layers of the logarithms, we conclude that the domain of \( f(x) = \ln(\ln(\ln x)) \) is all \( x \) such that \( x > e \).

Step 5: Attempt to Find the Inverse Function

To find the inverse \( f^{-1}(x) \), solve \( y = \ln(\ln(\ln x)) \) for \( x \) in terms of \( y \).

Step 6: Exponentiate to Undo Logarithms

Use exponentiation to undo the \( \ln \) functions: start by setting \( e^y = \ln(\ln x) \), then \( e^{e^y} = \ln x \), and \( e^{e^{e^y}} = x \).

Step 7: Write the Inverse Function

The inverse function is \( f^{-1}(x) = e^{e^{e^x}} \).

Key Concepts

Domain of a Function

The domain of a function represents all the possible input values for which the function is defined. To find the domain, we often need to consider the type of mathematical operations involved.When it comes to logarithmic functions, like the natural logarithm \( \ln(x) \), the domain is constrained to positive numbers. The natural logarithm, written as \( \ln(x) \), is only defined when \( x > 0 \) because you cannot take the logarithm of zero or a negative number in the real number system.For composite functions, like \( f(x) = \ln(\ln(\ln x)) \), determining the domain requires examining each layer:

• The innermost function \( \ln x \) requires \( x > 0 \).
• The secondary layer \( \ln(\ln x) \) demands \( \ln x > 0 \), meaning \( x > e^0 = 1 \).
• The outermost layer \( \ln(\ln(\ln x)) \) necessitates \( \ln(\ln x) > 0 \), implying \( \ln x > 1 \) or \( x > e \).

Thus, for \( f(x) \), the domain is \( x > e \), because all conditions must be satisfied simultaneously.

Exponential Functions

Exponential functions are mathematical functions of the form \( f(x) = a^x \), where \( a \) is a positive constant. They are characterized by the rate at which they increase or decrease, and they have unique properties that make them useful in various fields of science and mathematics.An important feature of exponential functions is that they are the inverses of logarithm functions. This means they "undo" the effect of logarithms and vice versa. For example, when solving for an inverse function, exponentiation is often used to cancel out a natural logarithm.
In the given exercise, the inverse function of \( f(x) = \ln(\ln(\ln x)) \) was found by exponentiating sequentially.To reverse the operations by layers:

• Start with the innermost equation: \( e^y = \ln(\ln x) \).
• Next, move outward: \( e^{e^y} = \ln x \).
• Finally, arrive at the solution: \( e^{e^{e^y}} = x \).

This demonstrates how each \( \ln \) is "undoed" by an exponential \( e^x \). The inverse function hence gives us \( f^{-1}(x) = e^{e^{e^x}} \).

Natural Logarithms

Natural logarithms are logarithms with a base of \( e \), where \( e \approx 2.718 \). The natural logarithm of a number \( x \), written as \( \ln(x) \), answers the question: "To what power must \( e \) be raised, to obtain \( x \)?" Natural logs have important applications in many domains, including the natural world, due to their unique growth behavior.In a mathematical context, natural logarithms:

• Are the inverse operations of exponential functions.
  This means you can use \( \ln \) to solve exponentiation equations: if \( e^y = x \), then \( y = \ln(x) \).
• Have a domain of \( x > 0 \), meaning they are only defined for positive real numbers.

The exercise highlights the building of composite logarithmic functions, showing that the largest inner value of the function sets off constraints for the outer layers. Solving problems involving layers, like \( \ln(\ln(\ln x)) \), involves unwrapping the outer \( \ln \)s by ensuring the input remains valid at each step.
This often requires checking conditions recursively, step by step through the layers, to find valid domains and resolve operations using their inverse—the exponentials.