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 [4(x + 1)] = log6 (4) + log6 (x + 1)

Verified step by step guidance

1. Step 1: Recall the properties of logarithms, specifically the product rule, which states that \( \log_b (MN) = \log_b (M) + \log_b (N) \).
2. Step 2: Apply the product rule to the left side of the equation \( \log_6 [4(x + 1)] \).
3. Step 3: Recognize that \( 4(x + 1) \) can be expressed as a product of 4 and \( (x + 1) \).
4. Step 4: According to the product rule, \( \log_6 [4(x + 1)] = \log_6 (4) + \log_6 (x + 1) \).
5. Step 5: Compare the transformed left side with the right side of the original equation to determine if they are equivalent.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that govern their behavior, including the product, quotient, and power rules. The product rule states that log_b(mn) = log_b(m) + log_b(n), which allows us to break down logarithmic expressions into simpler components. Understanding these properties is essential for manipulating and simplifying logarithmic equations.

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another, which can be useful for solving equations. It states that log_b(a) = log_k(a) / log_k(b) for any positive k. This concept is important when dealing with logarithmic equations that may not be easily solvable in their original form.

Equivalence of Logarithmic Expressions

Determining whether two logarithmic expressions are equivalent requires understanding how logarithmic identities work. For example, if two expressions can be simplified to the same form using logarithmic properties, they are equivalent. This concept is crucial for evaluating the truth of the given equation and making necessary adjustments if it is false.