Problem 140 - Thomas Calculus in SI Units

Question

Could \(x^{\ln 2}\) possibly be the same as \(2^{\ln x}\) for \(x>0\) ? Graph the two functions and explain what you see.

Step-by-Step Solution

Answer

Yes, the functions are equal and the graph confirms this as they overlap.

Step 1: Understand the Problem

We need to determine if the two expressions \(x^{\ln 2}\) and \(2^{\ln x}\) are equal for some positive \(x\). Then, graph these functions to provide visual evidence.

Step 2: Set the Equations Equal

Let's set the two expressions equal to each other: \(x^{\ln 2} = 2^{\ln x}\). Our task is to find if there's any \(x > 0\) that satisfies this equation.

Step 3: Apply Logarithms

Apply natural logarithms to both sides: \(\ln(x^{\ln 2}) = \ln(2^{\ln x})\). Using the logarithmic identity \(\ln(a^b) = b \ln a\), this becomes \(\ln 2 \cdot \ln x = \ln x \cdot \ln 2\). Thus, both sides are equal for any \(x > 0\).

Step 4: Graph the Functions

Graph the functions \(f(x) = x^{\ln 2}\) and \(g(x) = 2^{\ln x}\) using a graphing tool. Both functions should appear as the same curve for all \(x > 0\), confirming they are identical.

Step 5: Analyze the Graph

On the graph, you should see that the lines representing \(x^{\ln 2}\) and \(2^{\ln x}\) completely overlap each other when plotted across \(x > 0\). This visually confirms that \(x^{\ln 2}\) is indeed equal to \(2^{\ln x}\) for these values of \(x\).

Key Concepts

Logarithms

Logarithms are a fascinating concept in mathematics, especially useful when dealing with exponential equations and their solutions. A logarithm answers the question: "to what exponent must we raise a specific base to obtain a certain number?" As a result, it is often written in the form of \(\log_b(a) = c \), meaning "b raised to the power of c equals a".

• The natural logarithm, denoted as \( \ln \), uses the constant e (approximately 2.718) as its base. It is especially prevalent in calculus and complex problem solving.
• An important property of logarithms at work in this exercise is that \( \ln(a^b) = b \ln a \). This property allows us to manipulate exponents inside logarithmic expressions, making them easier to solve or compare.

In solving equations like \( x^{\ln 2} = 2^{\ln x} \), applying logarithms on both sides helps simplify the equation. You can see how using the identity \( \ln(a^b) = b \ln a \) makes it clear that both expressions are inherently the same regardless of the specific value of \( x \) as long as \( x > 0 \).

Graphing Functions

Graphing functions is a practical way to visually explore and verify relationships between different equations. It allows us to observe the behavior of functions across a range of values for their variables.

• Each function is represented as a curve or a line on a coordinate plane, with the x-axis typically representing the input values and the y-axis representing the output values of the function.
• By graphing \( f(x) = x^{\ln 2} \) and \( g(x) = 2^{\ln x} \), we can confirm if they produce identical outputs for the same input values.
• In this case, our graph showed that the two functions overlapped exactly, confirming they are equivalent for all \( x > 0 \).

By using graphical analysis, we not only rely on algebraic manipulation and properties but also have a visual proof that enhances our understanding and confidence in the mathematical relationships we uncover.

Exponential Functions

Exponential functions are a central concept in many areas of mathematics. They have the general form \( y = a^{x} \), where \( a \) is a constant base, and \( x \) is the exponent or variable. These functions are powerful and appear in expressions that grow rapidly.

• In the specific context of the problem, we are dealing with functions like \( x^{\ln 2} \) and \( 2^{\ln x} \). While these may seem different at first glance, logarithmic properties help us analyze and compare them.
• The transformation feature of exponential expressions is crucial. Using the property of logarithms, we can equate different-looking squared terms to find true relationships.
• Exponential functions often describe real-world phenomena such as population growth, radioactive decay, and compound interest. Understanding exponential functions' behavior can help predict and model such situations effectively.

In conclusion, exponential functions, along with logarithms, form an interconnected part of mathematical analysis that enables the tackling of complex questions like the equivalency of \( x^{\ln 2} \) and \( 2^{\ln x} \).