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

$\frac{\log_{b} x}{\log_{b} y} = \log_{b} x - \log_{b} y$

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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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주요 개념

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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.

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.

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.

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