Textbook Question
In Exercises 41–44:
a. Find f⁻¹(x).
43. f(x) = 5 − 4x, a = 1/2
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.8.5a
5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
a. log_3(x)
Verified step by step guidance1
Recall that logarithms with different bases differ only by a constant multiple. Specifically, for any base \(a > 0, a \neq 1\), \(\log_a(x) = \frac{\ln(x)}{\ln(a)}\).
Since \(\log_3(x) = \frac{\ln(x)}{\ln(3)}\), it is exactly \(\ln(x)\) multiplied by a positive constant \(\frac{1}{\ln(3)}\).
When comparing growth rates as \(x \to \infty\), multiplying by a positive constant does not change the rate of growth, so \(\log_3(x)\) and \(\ln(x)\) grow at the same rate.
Therefore, \(\log_3(x)\) neither grows faster nor slower than \(\ln(x)\); they grow at the same rate asymptotically.
In summary, \(\log_3(x)\) grows at the same rate as \(\ln(x)\) as \(x \to \infty\).
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Growth Rates of Logarithmic Functions
Logarithmic functions with different bases differ only by a constant factor. For example, log base 3 of x can be expressed as ln(x) divided by ln(3). This means all logarithmic functions grow at the same rate asymptotically, differing only by a constant multiplier.
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Intro To Related Rates
Asymptotic Behavior and Limits
As x approaches infinity, comparing growth rates involves analyzing limits of ratios of functions. If the limit of f(x)/g(x) is a positive finite constant, f and g grow at the same rate. If the limit is zero or infinity, one grows slower or faster respectively.
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Big-O and Little-o Notation
Big-O and little-o notation describe relative growth rates of functions. If f(x) = O(g(x)), f grows no faster than g asymptotically. If f(x) = o(g(x)), f grows strictly slower than g. These notations help classify functions like logarithms, polynomials, and exponentials by their growth.
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Related Practice
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
5. a. arccos(1/2)
Textbook Question
2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
a. 10x^4 + 30x + 1
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Textbook Question
Find the inverse of f(x)=-x+1. Graph the line y=-x+1 together with the line y=x. At what angle do the lines intersect?
Textbook Question
a. Show that h(x) = x³ / 4 and k(x) = (4x)^(1/3) are inverses of one another.
Textbook Question
77. Which one is correct, and which one is wrong? Give reasons for your answers.
a. lim (x → 3) (x - 3) / (x² - 3) = lim (x → 3) 1 / (2x) = 1/6