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. [log(x + 2)/log(x - 1)] = log(x + 2) - log(x - 1)

Key Concepts

Logarithmic Properties

Logarithmic properties are rules that govern the manipulation of logarithms. Key properties include the product rule (log(a*b) = log(a) + log(b)), the quotient rule (log(a/b) = log(a) - log(b)), and the power rule (log(a^b) = b*log(a)). Understanding these properties is essential for simplifying logarithmic expressions and solving equations involving logarithms.

Change of Base Formula

The change of base formula allows the conversion of logarithms from one base to another, expressed as log_b(a) = log_k(a) / log_k(b) for any positive k. This is particularly useful when dealing with logarithms that are not easily simplified or when using calculators that typically only compute logarithms in base 10 or e. Mastery of this formula aids in verifying the truth of logarithmic equations.

Domain of Logarithmic Functions

The domain of a logarithmic function is restricted to positive real numbers. For the expression log(x + 2) and log(x - 1) to be defined, the arguments (x + 2) and (x - 1) must be greater than zero, leading to the conditions x > -2 and x > 1, respectively. Understanding these domain restrictions is crucial for determining the validity of logarithmic equations and ensuring that all operations are mathematically sound.