Textbook Question
Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.
ƒ(x) = |2x + 1|
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Ch. 1 - Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 1, Problem 1.20
For a certain constant a>1, ln a≈3.8067 . Find approximate values of log₂ a and logₐ 2 using the fact that ln 2≈0.6931.
Verified step by step guidance1
Use the change of base formula for logarithms: \( \log_b a = \frac{\ln a}{\ln b} \).
To find \( \log_2 a \), apply the formula: \( \log_2 a = \frac{\ln a}{\ln 2} \).
Substitute the given values: \( \ln a \approx 3.8067 \) and \( \ln 2 \approx 0.6931 \).
Calculate \( \log_2 a \) using the substituted values: \( \log_2 a = \frac{3.8067}{0.6931} \).
To find \( \log_a 2 \), use the reciprocal property of logarithms: \( \log_a 2 = \frac{1}{\log_2 a} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Change of Base Formula
The change of base formula allows us to convert logarithms from one base to another. It states that log_b(x) = log_k(x) / log_k(b) for any positive k. This is particularly useful when we need to compute logarithms in a base that is not readily available, such as converting natural logarithms (ln) to base 2 or base a.
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Change of Base Property
Natural Logarithm (ln)
The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is a fundamental concept in calculus and is used extensively in various applications, including growth models and compound interest. In this problem, ln a and ln 2 are provided, which are essential for calculating log₂ a and logₐ 2.
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Approximation of Logarithmic Values
In this context, approximating logarithmic values involves using known logarithmic values to estimate others. Given ln a and ln 2, we can derive log₂ a and logₐ 2 using the change of base formula. This method is particularly useful when exact values are difficult to compute, allowing for practical estimations in mathematical problems.
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Related Practice
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Missing piece Let g(x) = x² + 3 Find a function ƒ that produces the given composition.
(g o ƒ ) (x) = x⁴ + 3
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Simplify the difference quotient ƒ(x+h)-ƒ(x)/h
ƒ(x) = (x)/(x+1)
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Textbook Question
Convert the following expressions to the indicated base.
using base e
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