Problem 76 - Algebra and Trigonometry

Question

\(75-76\) 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

Domain: \(x > e^e\). Inverse: \(f^{-1}(y) = e^{e^{e^y}}\).

Step 1: Understand the Domain of the Function

The function given is \(f(x) = \ln (\ln (\ln x))\). For each natural logarithm (\(\ln\)) function to be defined, its argument must be greater than 0. Therefore, we set up the conditions step by step.To have \(\ln x\) defined, \(x > 1\). For \(\ln (\ln x)\) to be defined, it requires \(\ln x > 0\), which implies \(x > e\). For \(\ln (\ln (\ln x))\) to be defined, \(\ln (\ln x) > 0\), leading to \(\ln x > e\) or \(x > e^e\).Thus, the domain of \(f(x)\) is \(x > e^e\).

Step 2: Set Up the Inverse Function Problem

To find the inverse, start by setting \(y = f(x) = \ln (\ln (\ln x))\). Our goal is to express \(x\) in terms of \(y\). This would result in the inverse function, \(f^{-1}(x)\).

Step 3: Solve for \(x\) in Terms of \(y\)

Starting with \(y = \ln (\ln (\ln x))\), we reverse the operations. Begin by applying the exponential function to both sides: \[ e^y = \ln (\ln x) \]Next, apply the exponential function again:\[ e^{e^y} = \ln x \]Finally, apply the exponential function one more time to find \(x\):\[ x = e^{e^{e^y}} \]

Step 4: Express the Inverse Function

From the expression obtained, \(x = e^{e^{e^y}}\), we identify this as the inverse function of \(f\). Hence, the inverse function is:\[ f^{-1}(y) = e^{e^{e^y}} \]

Key Concepts

Domain of a function

The domain of a function refers to the set of all possible input values (usually represented by \( x \)) for which the function is defined. For the function \( f(x) = \ln (\ln (\ln x)) \), determining the domain involves a step-by-step consideration of where each nested natural logarithm remains valid.
First, remember that the natural logarithm, \( \ln \), is only defined for positive numbers. This means for \( \ln x \) to be defined, \( x \) must be greater than 1. But there’s more, for the next layer, \( \ln (\ln x) \), \( \ln x \) itself must be positive, which limits \( x \) to values greater than \( e \) (since \( \ln(e) = 1 \) and \( 1 > 0 \)).
Finally, the innermost function \( \ln(\ln(\ln x)) \) requires \( \ln(\ln x) > 0 \), pushing the required \( \ln x \) value further to be greater than \( e \), thus \( x > e^e \). Here's the step-by-step breakdown:

• For \( \ln x \): \( x > 1 \)
• Then for \( \ln (\ln x) \): \( x > e \)
• Finally for \( \ln (\ln (\ln x)) \): \( x > e^e \)

Hence, the domain of \( f(x) \) is \( x > e^e \). This ensures all the nested logarithmic operations result in defined and valid outputs.

Natural logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \) is an irrational and transcendental constant approximately equal to 2.71828. It is a fundamental operation in calculus and mathematical analysis. The natural logarithm describes continuous growth processes and decays in fields ranging from finance to biology.
The operation of \( \ln(x) \) essentially answers: "To what power must \( e \) be raised, to produce \( x \)?" For example, if \( x = e \), then \( \ln(e) = 1 \) because \( e^1 = e \).
Due to its unique properties, the natural logarithm is often encountered in solving problems involving exponential growth or decay, complex numbers, and even in computing compound interest. Additionally, it notably transforms multiplicative processes into additive ones, simplifying problem-solving in various contexts.

• \( \ln(xy) = \ln(x) + \ln(y) \)
• \( \ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y) \)
• \( \ln(x^a) = a \cdot \ln(x) \)

The properties of the natural logarithm make it pivotal in simplifying and solving equations, particularly when solving for inverse functions as outlined in the provided exercise.

Exponential functions

Exponential functions are functions of the form \( f(x) = a^x \), where the base \( a \) is a positive constant, and the exponent \( x \) is a variable. One of the most common bases for exponential functions is \( e \), leading to the expression \( f(x) = e^x \), which naturally complements the natural logarithm \( \ln(x) \). These functions are key in modeling situations with constant relative growth rates, such as populations, investments, and radioactive decay.
The inverse relationship between exponential functions and natural logarithms is fundamental. For instance, if \( y = \ln(x) \), then \( x = e^y \). This powerful relationship allows transformations and inversions of logarithmic functions into linear ones, simplifying complex exponential growth or decay problems.

• The function \( e^x \) is its own derivative: \( \frac{d}{dx}e^x = e^x \)
• The integral of \( e^x \) is \( \int e^x \, dx = e^x + C \)
• Inverse: swaps input and output \( e^{\ln(x)} = x \)

Understanding these relationships can greatly enhance problem-solving skills, especially when working with transformations, like those required for finding inverse functions in exercises that involve nested logarithmic functions.