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.log6 [(x - 1)/(x^2 + 4)] = log6 (x - 1) - log6 (x^2 + 4)

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insert step 1: Recall the properties of logarithms, specifically the quotient rule, which states that \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \).

insert step 2: Identify the components in the given equation: \( M = x - 1 \) and \( N = x^2 + 4 \).

insert step 3: Apply the quotient rule to the left side of the equation: \( \log_6 \left( \frac{x - 1}{x^2 + 4} \right) = \log_6 (x - 1) - \log_6 (x^2 + 4) \).

insert step 4: Compare the transformed left side of the equation with the right side of the original equation.

insert step 5: Conclude that the equation is true because both sides are equivalent after applying the quotient rule.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Logarithms

Logarithms have specific properties that simplify expressions. One key property is the quotient rule, which states that the logarithm of a quotient is equal to the difference of the logarithms: log_b(a/c) = log_b(a) - log_b(c). Understanding this property is essential for manipulating logarithmic equations and verifying their validity.

Domain of Logarithmic Functions

The domain of a logarithmic function is restricted to positive values. For the expression log_b(x), x must be greater than zero. In the given equation, both (x - 1) and (x^2 + 4) must be positive for the logarithms to be defined, which impacts the validity of the equation and any transformations made.

Equivalence of Logarithmic Expressions

Two logarithmic expressions are equivalent if they represent the same value for all permissible inputs. To determine if the given equation is true or false, one must analyze whether the left-hand side equals the right-hand side under the constraints of their domains. If they are not equivalent, adjustments must be made to create a true statement.

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