Textbook Question
86. This exercise explores the difference between
lim(x→∞)(1 + 1/x²)^x
and
lim(x→∞)(1 + 1/x)^x = e
c. Confirm your estimate of lim(x→∞)f(x) by calculating it with l’Hôpital’s Rule.
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 7.2.4c
4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
c. 3ln ∛(t² - 1) - ln(t+1)
Verified step by step guidance1
Identify the properties of logarithms that will help combine the terms: the power rule \(a \ln b = \ln b^a\) and the subtraction rule \(\ln A - \ln B = \ln \left( \frac{A}{B} \right)\).
Apply the power rule to the first term: rewrite \(3 \ln \sqrt[3]{t^2 - 1}\) as \(\ln \left( \sqrt[3]{t^2 - 1} \right)^3\).
Simplify the expression inside the logarithm by raising the cube root to the third power: \(\left( \sqrt[3]{t^2 - 1} \right)^3 = t^2 - 1\).
Rewrite the expression now as \(\ln (t^2 - 1) - \ln (t + 1)\).
Use the subtraction rule to combine the logarithms into a single term: \(\ln \left( \frac{t^2 - 1}{t + 1} \right)\).
Verified video answer for a similar problem:
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Logarithms
Logarithmic properties include rules such as the product, quotient, and power rules. These allow combining or breaking down logarithmic expressions, for example, ln(a) + ln(b) = ln(ab), ln(a) - ln(b) = ln(a/b), and k·ln(a) = ln(a^k). Understanding these is essential to rewrite expressions as a single logarithm.
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Change of Base Property
Exponent and Root Relationships
Roots can be expressed as fractional exponents, such as the cube root of x being x^(1/3). This allows rewriting expressions inside logarithms with roots into powers, facilitating the use of logarithm power rules to simplify or combine terms.
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7:39Introduction to Exponent Rules
Combining Logarithmic Expressions
To write multiple logarithmic terms as a single term, apply the properties of logarithms step-by-step: first convert coefficients to exponents, then use product or quotient rules to combine terms. This process simplifies complex expressions into one logarithm.
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7:30Logarithms Introduction
Related Practice
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
7. c. arcsec(-2)
Textbook Question
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
c. x²e^(-x)
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 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
1. 2y' + 3y = e^(-x)
c. y = e^(-x) + Ce^(-(3/2)x)