Textbook Question
9. True, or false? As x→∞,
e. e^x = o(e^(2x))
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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 7.2.2f
2. Express the following logarithms in terms of ln 5 and ln 7.
f. (ln35 + ln(1/7))/(ln25)
Verified step by step guidance1
Recall the logarithm properties: \(\ln(ab) = \ln a + \ln b\) and \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\).
Rewrite \(\ln 35\) as \(\ln(5 \times 7)\), which becomes \(\ln 5 + \ln 7\) using the product rule.
Rewrite \(\ln \left(\frac{1}{7}\right)\) as \(\ln 1 - \ln 7\). Since \(\ln 1 = 0\), this simplifies to \(-\ln 7\).
Add the terms in the numerator: \((\ln 35 + \ln(1/7)) = (\ln 5 + \ln 7) + (-\ln 7) = \ln 5\).
Rewrite the denominator \(\ln 25\) as \(\ln(5^2) = 2 \ln 5\). Then express the entire fraction as \(\frac{\ln 5}{2 \ln 5}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Logarithms
Logarithms have key properties such as the product rule (ln a + ln b = ln(ab)), the quotient rule (ln a - ln b = ln(a/b)), and the power rule (ln(a^b) = b ln a). These allow complex logarithmic expressions to be simplified or rewritten in terms of simpler logarithms.
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05:36
Change of Base Property
Change of Base and Logarithm Simplification
Simplifying logarithmic expressions often involves rewriting numbers as products or powers of known bases. For example, expressing 35 as 5 × 7 or 25 as 5² helps rewrite logarithms in terms of ln 5 and ln 7, facilitating easier manipulation and evaluation.
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05:36Change of Base Property
Natural Logarithm (ln) and Its Use
The natural logarithm, denoted ln, is the logarithm with base e. It is commonly used in calculus and higher mathematics. Understanding how to express other logarithms in terms of ln 5 and ln 7 requires familiarity with ln’s properties and how to decompose numbers into factors involving 5 and 7.
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05:18Derivative of the Natural Logarithmic Function
Related Practice
Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
f. lim(x→∞) coth x
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Textbook Question
1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
f. (e^x)/2
Textbook Question
112. True, or false? Give reasons for your answers.
e. sec^(-1)x = O(1)
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:
e. Plot the functions f and g, the identity, the two tangent lines, and the line segment joining the points (x_0, f(x_0)) and (f(x_0), x_0). Discuss the symmetries you see across the main diagonal y=x.
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
Textbook Question
1. Express the following logarithms in terms of ln 2 and ln 3.
f. ln √(13.5)