Problem 22 - Precalculus

Question

Solve the equation analytically. $$ \ln (\ln (x))=3 $$

Step-by-Step Solution

Answer

The solution is \( x = e^{e^3} \).

Step 1: Understand the Equation

We are given the equation \( \ln(\ln(x)) = 3 \). Our goal is to find the value of \( x \) that satisfies this equation.

Step 2: Exponentiate to Remove the Outer Logarithm

To eliminate the outer logarithm, we exponentiate both sides of the equation. This gives us \( \ln(x) = e^3 \) because the exponential function is the inverse of the natural logarithm.

Step 3: Exponentiate Again to Solve for x

Now that we have \( \ln(x) = e^3 \), we exponentiate again to solve for \( x \). Doing this results in \( x = e^{e^3} \).

Key Concepts

Exponential Function

The exponential function is a significant mathematical concept that is central to solving many types of equations, including logarithmic equations. In this context, the exponential function is often written as \( e^y \), where \( e \) is Euler's number, approximately equal to 2.718.

This function is so essential because it

• grows exceedingly fast,
• is the inverse of the natural logarithm,
• appears in numerous real-world applications such as compound interest and population growth.

When dealing with equations involving natural logarithms, like the one in our exercise, the exponential function becomes a powerful tool. Here, we use it to undo or "reverse" the logarithmic part of the equation. In our problem, after determining that \( \ln(x) = e^3 \), we apply the exponential function again by calculating \( x = e^{e^3} \). This helps us isolate \( x \) and ultimately solve the equation.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the inverse function of the exponential function \( e^x \). It is particularly valued in calculations involving growth and decay models.

The natural logarithm has several key properties:

• It is only defined for positive values of \( x \).
• The natural logarithm turns multiplication into addition, known as its property: \( \ln(ab) = \ln(a) + \ln(b) \).
• It simplifies exponentiation: \( \ln(e^x) = x \).

In our exercise, we start with the equation \( \ln(\ln(x))=3 \), where the objective was first to address the "outer" natural logarithm. By applying the inverse property with the exponential function, we transformed the equation into \( \ln(x) = e^3 \). This step strategically peeled away one layer of complexity, allowing us to focus on solving for \( x \) in the next step.

Inverse Functions

Inverse functions are incredibly useful in mathematics, acting as mirrors to "undo" operations carried out by the original function. For a function \( f(x) \), its inverse \( f^{-1}(x) \) is such that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

They are commonly used in:

• Transforming equations into simpler forms.
• Solving equations where one function dominates.
• Switching from a complex expression into a manageable one.

In our specific problem, the natural logarithm and the exponential function exemplify inverse relationships. The log equation \( \ln(\ln(x)) = 3 \) was simplified by recognizing that \( e^{\ln(a)} = a \) and \( \ln(e^b) = b \). By exponentiating each step, we methodically reversed the logarithmic layers until we isolated \( x \) as \( e^{e^3} \). Thus, understanding inverse functions is crucial for manipulating and solving equations efficiently.