Textbook Question
Solve each equation. Round answers to the nearest hundredth as needed. x2/3 =36
Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 5, Problem 8
Answer each of the following. Between what two consecutive integers must log2 12 lie?
Verified step by step guidance1
Recall that \( \log_2 12 \) means the exponent to which 2 must be raised to get 12.
Identify powers of 2 that are close to 12. For example, \( 2^3 = 8 \) and \( 2^4 = 16 \).
Since 12 is between 8 and 16, \( \log_2 12 \) must be between 3 and 4 because \( 2^3 < 12 < 2^4 \).
Therefore, the two consecutive integers between which \( \log_2 12 \) lies are 3 and 4.
This means \( 3 < \log_2 12 < 4 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithms and Their Meaning
A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log₂12 asks for the power to which 2 must be raised to get 12. Understanding this helps in estimating or calculating logarithmic values.
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Logarithms Introduction
Properties of Logarithms
Logarithms have properties such as log_b(mn) = log_b(m) + log_b(n), which can simplify calculations. Recognizing that 12 lies between powers of 2 (like 8 and 16) helps estimate log₂12 by comparing it to known integer logarithms.
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5:36Change of Base Property
Estimating Logarithmic Values Using Consecutive Integers
To find between which two integers a logarithm lies, identify consecutive powers of the base that bracket the number. Since 2³ = 8 and 2⁴ = 16, and 12 is between 8 and 16, log₂12 must lie between 3 and 4.
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5:26Graphs of Logarithmic Functions
Related Practice
Textbook Question
Solve each equation. Round answers to the nearest hundredth as needed. (1/4)x=64
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Textbook Question
Determine whether each function graphed or defined is one-to-one.
Textbook Question
Find each value. If applicable, give an approximation to four decimal places. log 1012
Textbook Question
Fill in the blank(s) to correctly complete each sentence. The graph of ƒ(x) = -(1/3)x+4-5 is that of ƒ(x) = (1/3)x reflected across the ______ -axis, translated to the left ______ units and down _______ units.
Textbook Question
Answer each of the following. Write log3 12 in terms of natural logarithms using the change-of-base theorem.