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. log4 (2x^3) = 3 log4 (2x)

Key Concepts

Logarithmic Properties

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

Change of Base Formula

The change of base formula allows you to convert 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.

Equivalence of Logarithmic Expressions

Equivalence of logarithmic expressions involves determining whether two logarithmic statements represent the same value. This requires understanding how to manipulate and simplify logarithmic expressions using properties and rules. In the context of the given equation, verifying equivalence will involve applying logarithmic properties to both sides to see if they yield the same result, thus establishing whether the equation is true or false.