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The Product Rule for Logarithms is a fundamental concept in logarithmic algebra that allows us to simplify logarithmic expressions involving multiplication. The rule states that for any positive numbers \(X\) and \(Y\), and a logarithm with base \(b\), the following holds:

• \( \log_b(XY) = \log_b(X) + \log_b(Y) \)

This means that the logarithm of a product is the sum of the logarithms.
In our original problem, we had \( \log_2(AB^2) \), where you can identify \( A \) and \( B^2 \) as factors.
Applying the product rule, the expression can be broken down into two log terms: \( \log_2(A) + \log_2(B^2) \).
This simplification helps in making complex logarithmic expressions more manageable by treating each factor separately.

The Power Rule for Logarithms is another essential tool in dealing with logarithmic expressions, especially where exponents are involved. According to this rule, the logarithm of a number raised to a power can be simplified using the formula:

• \( \log_b(X^k) = k \log_b (X) \)

Here, \(k\) can be any real number, and \(X\) is a positive number. This rule is particularly useful for breaking down expressions with exponents.
In our step-by-step solution, we encounter the term \( \log_2(B^2) \).
Applying the power rule, this expression is simplified to \( 2 \log_2(B) \), since the exponent \(2\) moves to the front of the logarithm.
This transformation makes further calculations easier by reducing an exponentiation to a multiplication.

Logarithmic Expansion refers to the process of breaking down a single complicated logarithmic expression into simpler components. Typically, this involves using the Product Rule, Power Rule, and Quotient Rule of logarithms. By expanding an expression, you can see its internal structure more clearly, which can be crucial for solving equations or further mathematical manipulation.
In the context of our problem, expanding \( \log_2(AB^2) \) results in the simpler form \( \log_2(A) + 2 \log_2(B) \).

• First, we applied the Product Rule to separate the terms involving \(A\) and \(B\).
• Next, the Power Rule simplified \(B^2\) into \(2 \log_2(B)\).

The result, \( \log_2(A) + 2 \log_2(B) \), is much simpler and allows for more straightforward mathematical operations.
This method of expansion is a powerful technique to handle complex logarithmic equations effectively.