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)

Key Concepts

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.