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. Recall the logarithm property that states: \(\log_b (MN) = \log_b (M) + \log_b (N)\), where \(b\) is the base of the logarithm, and \(M\) and \(N\) are positive numbers.
2. Identify the terms inside the logarithm on the left side: \$4(x + 1)\( can be seen as the product of \(4\) and \)(x + 1)$.
3. Apply the logarithm property to the left side: \(\log_6 [4(x + 1)]\) should equal \(\log_6 (4) + \log_6 (x + 1)\) if the property holds.
4. Since the right side of the equation is exactly \(\log_6 (4) + \log_6 (x + 1)\), the original equation matches the logarithm product rule.
5. Conclude that the equation is true for all \(x\) where \$4(x + 1) > 0\(, which means \)x > -1$ to keep the logarithms defined.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the product rule: log_b(MN) = log_b(M) + log_b(N). This property allows the logarithm of a product to be expressed as the sum of logarithms, which is essential for verifying or manipulating logarithmic equations.

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(A) requires that the argument A be positive (A > 0). Understanding this is crucial to determine if expressions like log6(4(x + 1)) and log6(x + 1) are defined, which affects the validity of the equation.

Equation Verification and Transformation

To verify if a logarithmic equation is true, one must apply logarithmic properties correctly and check for equivalence. If false, adjusting the equation by applying correct properties or rewriting terms ensures the statement becomes true, reinforcing understanding of algebraic manipulation.