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The Product Rule of Logarithms is a powerful tool for breaking down complex logarithmic expressions. When you have a logarithm of a product, like \( \log_b(xy) \), the rule allows you to expand this into two separate logarithms: \( \log_b x + \log_b y \).

This is especially helpful when you want to simplify or work with logarithmic expressions that include products within them. In the exercise, the expression \( \log_2(2x) \) is identified as a product of 2 and \( x \). By applying the product rule, it becomes \( \log_2 2 + \log_2 x \).

Understanding this rule can make dealing with logarithmic expressions more manageable and less intimidating. Always remember that breaking down problems into smaller parts can ease your way through solving them.

Simplification involves transforming expressions into their most basic form. This is not only to make them easier to understand but also to work with them efficiently. In the context of logarithms, simplifying sometimes means applying rules and properties to express the equation in its simplest terms.

In the example given, after applying the product rule, we had \( \log_2 2 + \log_2 x \). Simplification comes in by recognizing that \( \log_2 2 \) equals 1. Why? Because 2 raised to the power of 1 is 2, or in other words, \( 2^1 = 2 \). Thus, we can simplify \( \log_2 2 \) directly to 1.

So, the original expression ultimately becomes \( 1 + \log_2 x \). This basic form is easy to compute and understand, as all unnecessary complexities have been minimized.

Logarithmic expressions involve logarithms, which are the inverse operations of exponentiation. They allow us to solve problems involving exponential growth or decay by transforming multiplicative problems into additive ones through the use of logarithmic rules.

In the context of the exercise \( \log_2(2x) \), we faced a common situation where a logarithm contains a product, making it necessary to apply logarithmic rules, such as the product rule, to simplify the expression.

Logarithmic expressions can initially seem complex, but by understanding and utilizing the laws of logarithms like the product, quotient, and power rules, they can be expanded into simpler, more manageable forms. This not only aids in computation but also in gaining a deeper understanding of how logarithms function in various mathematical contexts.