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. $\left[\frac{log(x+2)}{log(x-1)}\right]=log(x+2)-log(x-1)$

Verified step by step guidance

1. 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 \( \log(a) - \log(b) \).
2. 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.
3. 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.
4. 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.
5. 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.

Key Concepts

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.