In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $[\frac{l o g (x + 2)}{l o g (x - 1)}] = l o g (x + 2) - l o g (x - 1)$

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Recall the properties of logarithms: the quotient rule states that $ \log(a) - \log(b) = \log\left(\frac{a}{b}\right) $, and the division of logarithms $ \frac{\log(a)}{\log(b)} $ is not equivalent to $ \log(a) - \log(b) $.

Examine the given equation: $ \frac{\log(x + 2)}{\log(x - 1)} = \log(x + 2) - \log(x - 1) $. Notice that the left side is a division of two logarithms, while the right side is a difference of two logarithms.

Apply the logarithm difference property to the right side: $ \log(x + 2) - \log(x - 1) = \log\left(\frac{x + 2}{x - 1}\right) $. This shows the right side is a single logarithm of a quotient.

Since $ \frac{\log(x + 2)}{\log(x - 1)} $ is not generally equal to $ \log\left(\frac{x + 2}{x - 1}\right) $, the original equation is false.

To make the statement true, replace the left side with $ \log(x + 2) - \log(x - 1) $ or replace the right side with $ \frac{\log(x + 2)}{\log(x - 1)} $ depending on the intended meaning.

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주요 개념

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Properties of Logarithms

Logarithmic properties include rules for simplifying expressions, such as the product, quotient, and power rules. For example, log(a) - log(b) equals log(a/b), but a quotient of logs like log(a)/log(b) does not simplify to a difference. Understanding these properties helps determine if the given equation is true.

Difference Between Logarithm Quotient and Quotient of Logarithms

The quotient rule for logarithms states log(a) - log(b) = log(a/b), which is different from dividing two logarithms, log(a)/log(b). The expression log(x + 2)/log(x - 1) is a ratio of two logs, not a log of a quotient, so it cannot be simplified to a difference of logs.

Domain Restrictions for Logarithmic Functions

Logarithms are defined only for positive arguments, so x + 2 > 0 and x - 1 > 0 must hold. Identifying these domain restrictions is essential to ensure the expressions are valid and to avoid undefined values when evaluating or modifying the equation.