Textbook Question
What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line y=x?
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 7.2.4c
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: \(\ln \left( \frac{t^2 - 1}{t + 1} \right)\).
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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 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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Guided course
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
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
3. c. sin^(-1)(-√3/2)
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
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, solve for t.
1. c. e^((ln 0.2)t) = 0.4