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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 77b
Chapter 1, Problem 77b

Determine whether the following statements are true and give an explanation or counterexample.


logbxlogby=logbx−logby\(\frac{\log_{b}\)x}{\(\log\)_{b}y}=\(\log\)_{b}x-\(\log\)_{b}y

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1
Consider the properties of logarithms, specifically the quotient rule: \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \).
The given statement is \( \frac{\log_b x}{\log_b y} = \log_b x - \log_b y \).
To verify, let's substitute \( x = b^m \) and \( y = b^n \), where \( m \) and \( n \) are real numbers.
Calculate \( \log_b x = m \) and \( \log_b y = n \), then \( \frac{\log_b x}{\log_b y} = \frac{m}{n} \).
Compare \( \frac{m}{n} \) with \( m - n \). If they are not equal, the statement is false.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logarithmic Properties

Logarithms have specific properties that govern their behavior, including the quotient rule, which states that the logarithm of a quotient is the difference of the logarithms. This means that for any positive numbers x and y, the equation log_b(x/y) = log_b(x) - log_b(y) holds true. Understanding these properties is essential for manipulating logarithmic expressions correctly.
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Change of Base Property

Change of Base Formula

The change of base formula allows us to express logarithms in terms of logarithms of a different base. Specifically, log_b(x) can be rewritten as log_k(x) / log_k(b) for any positive k. This concept is crucial when comparing logarithms of different bases and can help simplify complex logarithmic expressions.
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Change of Base Property

Counterexamples in Mathematics

A counterexample is a specific case that disproves a general statement or conjecture. In the context of the given question, providing a counterexample would involve finding specific values of x and y that demonstrate the falsity of the statement. This concept is important in mathematical reasoning, as it helps validate or invalidate claims through concrete evidence.
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Slopes of Tangent Lines
Related Practice
Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


2=ln2e2=\(\ln\)2^{e}

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Textbook Question

A GPS device tracks the elevation EE (in feet) of a hiker walking in the mountains. The elevation tt hours after beginning the hike is given in the figure. <IMAGE>

Notice that the curve in the figure is horizontal for an interval of time near t=5.5t=5.5 hr. Give a plausible explanation for the horizontal line segment.

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

log546=4log56\(\log\)_54^6=4\(\log\)_56

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Textbook Question

In each exercise, a function and an interval of its independent variable are given. The endpoints of the interval are associated with points PP and QQ on the graph of the function.

a. Sketch a graph of the function and the secant line through PP and QQ.

b. Find the slope of the secant line in part (a), and interpret your answer in terms of an average rate of change over the interval. Include units in your answer.

After tt seconds, an object dropped from rest falls a distance d=16t2d=16t^2, where dd is measured in feet and 2≤t≤52\(\leq{t}\]\leq{5}\).

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Textbook Question

Roots and powers Sketch a graph of the given pairs of functions. Be sure to draw the graphs accurately relative to each other.


y = (x)¹⸍³ and y = (x)¹⸍⁵

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


2=10log1022=10^{\(\log\)_{10}2}

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