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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 77b
Capitolo 1, Problema 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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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
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