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The product rule of logarithms is a vital tool in simplifying logarithmic expressions. It allows us to break down complex log terms into simpler parts. The rule states: \[ \log_b(mn) = \log_b(m) + \log_b(n) \]
This means that for any given log with a product inside, you can separate it into the sum of the logs.
This is particularly useful when dealing with lengthy expressions.
For instance, if you have \log_{2}(3x)\, you can apply the product rule to get \[ \log_{2}(3x) = \log_{2}(3) + \log_{2}(x) \]
This simplification makes it easier to handle and work with logarithmic expressions.
The underlying idea is that the logarithm of a product is the sum of the logarithms of its factors.

Logarithmic expressions are equations that involve logarithms, which are the inverses of exponential functions.
A typical logarithmic expression looks like \log_{b}(x)\ where \(b\) is the base, and \(x\) is the argument.
Understanding how to manipulate these expressions is crucial in various areas of math.
There are different rules to manage these expressions, including:

• The Product Rule
• The Quotient Rule
• The Power Rule

In the exercise, the expression \log_{2}(3x)\ features a product within the logarithm.
Using the product rule, we transform this into simpler components:
\log_{2}(3) + \log_{2}(x)\
This way, you break down complex logarithms into manageable parts.
Such skills are important when solving equations or analyzing functions that encompass logs.

Logarithm rules are essential for simplifying and solving logarithmic equations. The main rules to remember include:

• Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
• Quotient Rule: \( \log_b\left( \frac{m}{n} \right) = \log_b(m) - \log_b(n) \)
• Power Rule: \( \log_b(m^n) = n \cdot \log_b(m) \)

These rules help to transform complicated logarithmic expressions into more straightforward forms.
In the given problem, the product rule was applied effectively:
\[ \log_{2}(3x) = \log_{2}(3) + \log_{2}(x) \]
This breakdown follows the product rule, turning a single log expression with a product inside into a sum of logs.
Mastering these rules aids in solving logarithmic equations and performing operations like addition, subtraction, and exponentiation within logs.
These skills are useful across different math topics, including algebra, calculus, and beyond.