Skip to main content
Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.3.155d
Chapter 7, Problem 7.3.155d

155. Which is bigger, πᵉ or e^π?
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.
d. Conclude that
xᵉ < eˣfor all positivex ≠ e.

Verified step by step guidance
1
Rewrite the expressions \( \pi^{e} \) and \( e^{\pi} \) in a form that allows comparison using logarithms. Consider taking the natural logarithm of both expressions to compare their sizes without directly calculating their values.
Express the comparison as \( \pi^{e} < e^{\pi} \) if and only if \( e \ln(\pi) < \pi \ln(e) \). Since \( \ln(e) = 1 \), this simplifies to comparing \( e \ln(\pi) \) and \( \pi \).
Define a function \( f(x) = \frac{\ln(x)}{x} \) for \( x > 0 \) to analyze the inequality \( x^{e} < e^{x} \) for \( x \neq e \). The inequality \( x^{e} < e^{x} \) is equivalent to \( e \ln(x) < x \), or \( \frac{\ln(x)}{x} < \frac{1}{e} \).
Find the critical points of \( f(x) = \frac{\ln(x)}{x} \) by differentiating: \( f'(x) = \frac{1 - \ln(x)}{x^{2}} \). Set \( f'(x) = 0 \) to find that the maximum occurs at \( x = e \).
Conclude that since \( f(x) \) attains its maximum at \( x = e \), for all positive \( x \neq e \), \( f(x) < f(e) = \frac{1}{e} \). Therefore, \( x^{e} < e^{x} \) for all positive \( x \neq e \), which includes the comparison between \( \pi^{e} \) and \( e^{\pi} \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential and Power Functions

Understanding the difference between expressions like πᵉ and e^π requires familiarity with power functions (x raised to a constant) and exponential functions (constant raised to variable). Recognizing how these functions behave for positive real numbers is essential to compare their values without direct computation.
Recommended video:
6:13
Exponential Functions

Function Comparison Using Logarithms

Comparing expressions like πᵉ and e^π can be simplified by taking natural logarithms, converting powers into products. This technique transforms the inequality into a comparison of products involving logarithms, making it easier to analyze without a calculator.
Recommended video:
5:26
Graphs of Logarithmic Functions

Monotonicity and Critical Points of the Function f(x) = x^{1/x}

The function f(x) = x^{1/x} reaches its maximum at x = e, which helps prove inequalities like xᵉ < eˣ for x ≠ e. Understanding how to find and interpret critical points and monotonicity of such functions is key to concluding the given inequality.
Recommended video:
04:56
Derivative of the Natural Exponential Function (e^x)
Related Practice
Textbook Question

1. Express the following logarithms in terms of ln 2 and ln 3.

d. ln ∛9

Textbook Question

86. This exercise explores the difference between

lim(x→∞)(1 + 1/x²)^x

and

lim(x→∞)(1 + 1/x)^x = e

c. Confirm your estimate of lim(x→∞)f(x) by calculating it with l’Hôpital’s Rule.

Textbook Question

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.

68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2

Textbook Question

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:


d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.


70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

1
views
Textbook Question

In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.

1. 2y' + 3y = e^(-x)

c. y = e^(-x) + Ce^(-(3/2)x)

Textbook Question

4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.

c. 3ln ∛(t² - 1) - ln(t+1)