Textbook Question
1. Express the following logarithms in terms of ln 2 and ln 3.
d. ln ∛9
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
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 guidance1
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 \( e^{\pi} \).
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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.
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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:
5:26Graphs 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.
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04:56Derivative of the Natural Exponential Function (e^x)
Related Practice
Textbook Question
2. Express the following logarithms in terms of ln 5 and ln 7.
d. ln 1225
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.
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3
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
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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.
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2