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When dealing with logarithms, understanding their properties can significantly simplify complex expressions. One such useful property is the **product property of logarithms**. This property states that the logarithm of a product of numbers is equivalent to the sum of the logarithms of the individual numbers.

This can be expressed as:

• For real numbers \(a\) and \(b\), and any positive base \(b\) (other than 1), the product property is: \( \log_b (a \cdot b) = \log_b a + \log_b b \).

This property is particularly useful when you encounter logarithms involving multiplicative expressions, such as \(\log(2 \cdot 3 \cdot x)\).

Example Application

For example, in our original exercise, the logarithmic expression \( \log(4x^2y) \) can be broken into simpler components. Applying the product property allows us to express this as \( \log 4 + \log x^2 + \log y \), effectively distributing the logarithm over the product into a sum.

Another critical property of logarithms is the **power property of logarithms**. This rule helps in dealing with logarithmic expressions that include terms with exponents. The basic rule here is:

"The logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the base number."

• Expressed mathematically: For real number \(a\) raised to the power \(n\), and a positive base \(b\), \( \log_b (a^n) = n \cdot \log_b a \).

Applying the Power Property

This property was employed in our step-by-step solution, where the term \( \log x^2 \) appeared. By using the power property, \( \log x^2 \) was rewritten as \( 2 \cdot \log x \). This makes solving such logarithmic equations more straightforward, by reducing the complexity caused by exponents.

**Logarithm expansion** refers to the process of breaking down a single logarithmic expression into multiple simpler logs that are easier to evaluate. It often involves applying both the product and power properties of logarithms.

Using Expansion Techniques

In our example, the expression \( \log(4x^2y) \) was expanded into \( \log 4 + 2\log x + \log y \). Breakdown steps include:

• Use the product property to split initial multiplication inside the log: \( \log(4) + \log(x^2) + \log(y) \).
• Apply the power property to rearrange logarithms with exponents: \( \log(4) + 2\log(x) + \log(y) \).
• Simplify any straightforward logarithms, like \( \log(4) \), according to the base. In this problem's context, viewing \( \log 4 \) in base 10 would lead us to consider it as a constant, 2.

Logarithm expansions are immensely helpful in algebra as they make large computations more manageable by breaking them into smaller, more basic parts.