Problem 143 - Thomas Calculus in SI Units

Question

Which is bigger, \(\pi^{e}\) or \(e^{\pi} ? \quad\) Calculators have taken some of the mystery out of this once-challenging question. (Go ahead and check; you will see that it is a very close call.) You can answer the question without a calculator, though. a. Find an equation for the line through the origin tangent to the graph of \(y=\ln x\) b. Give an argument based on the graphs of \(y=\ln x\) and the tangent line to explain why \(\ln x

Step-by-Step Solution

Answer

\(e^\pi > \pi^e\).

Step 1: Find the Slope of the Tangent Line

To find the equation of the tangent line through the origin to the graph of \(y = \ln x\), we first need to calculate the derivative of this function. The derivative \(y' = \frac{d}{dx}(\ln x) = \frac{1}{x}\). At \(x = e\), the slope of the tangent line is \(\frac{1}{e}\). Since the line passes through the origin, its equation is \(y = \frac{1}{e}x\).

Step 2: Compare the Log Function and Tangent Line

Now, let's compare the values of \(\ln x\) and \(\frac{x}{e}\). Since the graph of \(y = \ln x\) is below its tangent line at \(x = e\) for all \(x eq e\), it follows that \(\ln x < \frac{x}{e}\) for all positive \(x eq e\). This is because \(\ln x\) is concave down.

Step 3: Show Inequality for Logarithmic Powers

Using the result from Step 2, we have \(e \ln x < x\). This inequality implies \(\ln(x^e) = e\ln x < x\), showing that \(x^e < e^x\) for all positive \(x eq e\).

Step 4: Conclusion About the Original Problem

Since the inequality \(x^e < e^x\) holds true for all positive \(x eq e\), we can apply this to \(x = \pi\). Therefore, \(\pi^e < e^\pi\).

Step 5: Final Answer

We conclude that \(e^\pi\) is greater than \(\pi^e\). Thus, the answer to the question is \(e^\pi > \pi^e\).

Key Concepts

Derivative of logarithmic functions

When dealing with logarithmic functions, one core concept is understanding their derivatives. For the natural logarithm function, given by \( y = \ln x \), the derivative is \( y' = \frac{1}{x} \). This derivative tells us about the slope of the tangent line at any point on the curve of the logarithmic function.
In our exercise, we specifically calculate the slope at the point where \( x = e \). Here, the derivative simplifies to \( \frac{1}{e} \), indicating the slope of the tangent line passing through the origin. Understanding this slope is key to analyzing and comparing the behavior of the logarithmic curve against other functions, such as an exponential line.

Comparison of exponential expressions

Comparing exponential expressions like \( \pi^{e} \) and \( e^{\pi} \) might seem daunting at first glance. However, by leveraging the properties of logarithms and exponential functions, we can simplify and analyze the differences between them.
By using the inequality \( x^e < e^x \) for all positive \( x eq e \), derived from calculating derivatives and tangent lines, we can apply this comparison to specific values.
In this case, when \( x = \pi \), the inequality \( \pi^e < e^\pi \) holds, showing that \( e^{\pi} \) is indeed greater than \( \pi^{e} \).
This demonstrates how mathematical reasoning and fundamental properties of these functions help solve seemingly complex questions.

Graphical analysis of functions

Analyzing graphs is crucial in understanding inequalities involving logarithmic and exponential functions. The function \( y = \ln x \) is concave down, meaning it bends towards the x-axis. This graphical trait helps us conclude that \( \ln x < \frac{x}{e} \) for all \( x eq e \).
By examining the slope of the tangent line at \( x = e \), where its equation is \( y = \frac{1}{e} x \), we establish a clearer understanding of how the logarithm function behaves in comparison to linear functions.
The visual representation of these functions further solidifies that while the logarithmic curve is subtle in its increase, the line \( y = \frac{1}{e} x \) provides a point of contrast, helping to clarify the inequality \( \ln(x^e) = e\ln x < x \).
Therefore, graphical analysis supports our understanding of the relative size of exponential expressions, such as \( \pi^e \) and \( e^\pi \).

Mathematical problem solving

Solving mathematical problems often involves a mix of algebraic manipulation, graphical insight, and logical reasoning. In our exercise, we tackled the comparison between \( \pi^e \) and \( e^\pi \) through different approaches.

• We first found the derivative of the logarithmic function to help us understand the slope of the tangent line.
• Next, we compared logarithmic values and their tangent line equations to establish inequalities.
• Lastly, these inequalities helped us examine specific cases, like \( x = \pi \), to make a final conclusion.

These steps leverage multiple mathematical disciplines and concepts to weave through a problem efficiently. Each step builds upon the previous, leading us to a solid understanding of why \( e^\pi \) is indeed greater than \( \pi^e \). By taking calculated steps, comparing functions, and utilizing graphs, we enrich our problem-solving toolkit.