Textbook Question
91. [Technology Exercise] 91. The continuous extension of to (sin x)^x to [0, π]
b. Verify your conclusion in part (a) by finding lim(x→0⁺)f(x) with l’Hôpital’s Rule.
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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 7.2.1b
1. Express the following logarithms in terms of ln 2 and ln 3.
b. ln(4/9)
Verified step by step guidance1
Recall the logarithm property that allows you to express the logarithm of a quotient as the difference of logarithms: \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\).
Apply this property to the given expression: \(\ln\left(\frac{4}{9}\right) = \ln 4 - \ln 9\).
Express 4 and 9 in terms of their prime factors: \(4 = 2^2\) and \(9 = 3^2\).
Use the logarithm power rule, which states \(\ln(a^b) = b \ln a\), to rewrite \(\ln 4\) and \(\ln 9\) as \(2 \ln 2\) and \(2 \ln 3\) respectively.
Combine these results to express the original logarithm entirely in terms of \(\ln 2\) and \(\ln 3\): \(\ln\left(\frac{4}{9}\right) = 2 \ln 2 - 2 \ln 3\).
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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, quotient, and power rules. The quotient rule states that ln(a/b) = ln(a) - ln(b), allowing the expression of logarithms of fractions as differences of logarithms.
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05:36
Change of Base Property
Natural Logarithm (ln)
The natural logarithm, denoted ln, is the logarithm with base e. It is commonly used in calculus and can be manipulated using its properties to simplify expressions involving constants like 2 and 3.
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05:18Derivative of the Natural Logarithmic Function
Expressing Numbers as Powers or Products
To rewrite logarithms in terms of ln 2 and ln 3, numbers like 4 and 9 should be expressed as powers or products of 2 and 3, e.g., 4 = 2^2 and 9 = 3^2, enabling the use of the power rule for logarithms.
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04:21The Product Rule Example 1
Related Practice
Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
71. ∫(from 1/5 to 3/13)dx/(x√(1-16x²))
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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:
b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
4. b. arcsin(-1/√2)
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
2. b. tan^(-1)(√3)
Textbook Question
Suppose that the function f and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.
Assuming the inverse function f^(-1) is differentiable, find the slope of f^(-1)(x) at
b. x=2