Problem 148 - College Algebra

Question

Solve each equation. Check each proposed solution by direct substitution or with a graphing utility. $$ \ln (\ln x)=0 $$

Step-by-Step Solution

Answer

The solution to the equation \( \ln(\ln(x)) = 0 \) is \( x = e \)

Step 1: Solve the Equation

The equation given is \( \ln(\ln(x)) = 0 \). To solve for x, we must first get rid of the natural logarithm function. We know that \( e^0 = 1 \). Hence, we can raise both sides as the power of e, which gives us \( e^{\ln(\ln(x))} = e^0 \). From the properties of logarithms, this simplifies to \( \ln(x) = 1 \).

Step 2: Solve the Inner Function

To further solve for x, we will once again raise both sides as the power of e. So the equation becomes \( e^{\ln(x)} = e^1 \). Now, from the properties of logarithms, we get \( x = e \).

Step 3: Check the Solution

To verify the correctness of the solution, we can substitute the value of x back into the original equation. Substituting x = e in the equation yields \( \ln(\ln(e)) = 0 \). We know that \( \ln(e) = 1 \), so we have \( \ln(1) = 0 \) which confirms our solution is correct.

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln(x) \), is a special kind of logarithm that uses the number \( e \) as its base. The number \( e \), approximately 2.71828, is an irrational and transcendental number, important for its unique mathematical properties, especially in calculus and exponential growth. The natural logarithm tells you what power \( e \) must be raised to produce a given number. For example, \( \ln(e) = 1 \) because \( e^1 = e \). When you encounter equations involving \( \ln(x) \), such as \( \ln(\ln(x)) = 0 \), you can apply your understanding of the natural logarithm to simplify and solve the equation. Here, you recognize that solving \( \ln(a) = 0 \) typically means \( a = 1 \), as raising \( e \) to the power zero yields one.

Properties of Logarithms

Logarithms, including natural logarithms, have specific properties that simplify manipulation and problem-solving. One essential property is that the exponential function and the logarithmic function are inverses. This means that if you have \( e^{\ln(x)} \), it simplifies just to \( x \). This property is extremely useful for solving equations where variables are nested within logarithms, such as \( \ln(\ln(x)) \).

Another key property is the power rule: \( \ln(x^y) = y \cdot \ln(x) \), which helps when dealing with exponential expressions within a logarithm. Additionally, knowing that \( \ln(1) = 0 \) and \( \ln(e) = 1 \) provides fundamental values that aid in solving and checking equations. By comprehending these properties, you can more easily navigate through complex logarithmic equations by breaking them down step by step.

Substitution Method

The substitution method is a useful algebraic technique frequently applied to solve complicated equations. It involves replacing a complex operation or expression with a simpler placeholder and solving the resulting equation first. Once the simpler equation is solved, you substitute back to find the desired solution.

In the context of the exercise, you begin by recognizing \( \ln(\ln(x)) = 0 \) and simplify it by letting \( \ln(x) = y \). This reduces the equation to \( \ln(y) = 0 \), which implies \( y = 1 \). By substituting back, \( \ln(x) = 1 \), the problem becomes simpler and manageable, leading you to solve for \( x \). By using substitution, you effectively break down the problem and guide yourself through a logical pathway to find \( x = e \).

Exponential Functions

Exponential functions have formulas where variables appear in the exponent, commonly taking the form \( e^x \). The constant \( e \) is significant because it forms the base of the natural logarithm, creating a seamless connection between exponential and logarithmic operations. In equations like the one in the exercise \( \ln(\ln(x)) = 0 \), the exponential function helps in 'undoing' the natural log, thanks to their inverse nature.

When you solve \( \ln(x) = 1 \), exponential functions come to the rescue by allowing you to write \( x = e^{1} \), or simply \( x = e \). This inverse relationship is powerful, making exponential functions indispensable tools in solving equations involving logarithms. Understanding this interplay helps you tackle logarithmic equations with more confidence and clarity.