Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
73. ∫(from 0 to π)cos(x)dx/√(1+sin²x)
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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 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 7.2.3b
3. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
b. ln(3x² - 9x) + ln(1/3x)
Verified step by step guidance1
Recall the logarithm property that states: \(\ln(a) + \ln(b) = \ln(ab)\). This allows us to combine the sum of logarithms into a single logarithm of the product.
Identify the two expressions inside the logarithms: the first is \(3x^{2} - 9x\) and the second is \(\frac{1}{3}x\).
Multiply the two expressions inside the logarithms: \((3x^{2} - 9x) \times \left(\frac{1}{3}x\right)\).
Simplify the product by distributing and combining like terms: multiply \$3x^{2}\( by \(\frac{1}{3}x\) and \)-9x$ by \(\frac{1}{3}x\).
Write the final expression as a single logarithm: \(\ln\left(\text{simplified product}\right)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Logarithms
Logarithmic properties allow the simplification and combination of logarithmic expressions. Key properties include 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 rules help rewrite multiple logarithmic terms as a single logarithm.
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Change of Base Property
Domain of Logarithmic Functions
The domain of a logarithmic function consists of all positive real numbers inside the logarithm. When combining logarithms, it is essential to ensure the resulting argument remains positive, as ln(x) is undefined for x ≤ 0. This affects the validity of the simplified expression.
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5:26Graphs of Logarithmic Functions
Algebraic Simplification
Algebraic simplification involves factoring and reducing expressions inside the logarithms before applying logarithmic properties. For example, factoring 3x² - 9x as 3x(x - 3) can make it easier to combine terms and simplify the overall expression.
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05:25Determine Continuity Algebraically
Related Practice
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
c. x=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
75. b. Identify the function’s local and absolute extreme values, if any, saying where they occur.
g(x) = x(ln x)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:
c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
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:
b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.
68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2