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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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.

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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.

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Logarithms Introduction

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.

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Domanda del libro di testo

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. $$\log$_6 $\left$( $\frac{x - 1}{x^2 + 4}$ $\right$) = $\log$_6 (x - 1) - $\log$_6 (x^2 + 4)$