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 7.2.2d
2. Express the following logarithms in terms of ln 5 and ln 7.
d. ln 1225
Verified step by step guidance1
Start by factoring the number inside the logarithm to express it in terms of its prime factors or known bases. For 1225, find its prime factorization.
Recognize that 1225 can be written as a product of powers of 5 and 7, since 1225 = 35^2 and 35 = 5 \(\times\) 7.
Rewrite the logarithm using the property of logarithms that states \( \ln(a^b) = b \ln(a) \). So, \( \ln(1225) = \ln(35^2) = 2 \ln(35) \).
Next, use the logarithm product rule \( \ln(ab) = \ln(a) + \ln(b) \) to express \( \ln(35) \) as \( \ln(5) + \ln(7) \).
Combine all the steps to express \( \ln(1225) \) entirely in terms of \( \ln(5) \) and \( \ln(7) \) as \( 2 (\ln(5) + \ln(7)) \).
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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 such as the product, quotient, and power rules allow us to break down complex logarithms into simpler parts. For example, ln(ab) = ln a + ln b and ln(a^n) = n ln a. These rules help express logarithms in terms of known values.
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Change of Base Property
Prime Factorization
Prime factorization involves expressing a number as a product of prime numbers. This is useful in logarithms to rewrite the argument as a product of primes or their powers, enabling the use of logarithmic properties to simplify expressions.
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11:26Partial Fraction Decomposition: Repeated Linear Factors
Natural Logarithm (ln)
The natural logarithm, denoted ln, is the logarithm with base e. Understanding that ln is a specific logarithm function helps in manipulating and expressing logarithms in terms of known ln values, such as ln 5 and ln 7.
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Related Practice
Textbook Question
What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line y=x?
Textbook Question
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.
Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
c. lim(x→∞) sinh x
1
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Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
5. c. arccos(√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:
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