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The Product Rule is a fundamental property of logarithms that simplifies the addition of two logarithms with the same base. This rule states that adding two logs is equivalent to taking the logarithm of the product of their arguments. In mathematical terms, it can be expressed as:

• \( \log_b M + \log_b N = \log_b (M \cdot N) \)

This rule is extremely useful because it helps transform a sum of two logarithms into a simpler single logarithmic expression.
For example, if you have \( \log_{10} 30 + \log_{10} 2 \), by applying the Product Rule, you multiply 30 and 2 to get 60. Thus, the expression becomes \( \log_{10} 60 \).
Using the Product Rule means fewer terms and easier calculations for complex logarithmic equations.

Understanding the base of a logarithm is crucial when you work with logarithmic expressions. The base determines how many times you need to multiply the base to obtain the number you are taking the log of. Mathematically, it can be defined as follows:

• \( \log_b x \) answers the question, "What power do we need to raise \( b \) to, to obtain \( x \)?"

Common bases include 10 and \( e \) (the natural logarithm), but logs can have any positive value as a base. In the example, \( \log_{10} 30 \) and \( \log_{10} 2 \), the base 10 is often used in scientific calculations and is known as the common logarithm.
When combining logarithms, it is important to ensure that the bases of the logarithms are the same. Otherwise, the properties of logarithms, including the Product Rule, cannot be directly applied.

Combining logarithms is a process of simplifying multiple logarithmic terms into a single term. To do this effectively, similar bases and certain properties, such as the Product Rule, must be utilized. Here's how you can approach combining logarithms:

• Ensure the logarithms have the same base.
• Apply appropriate logarithmic rules, like the Product Rule or Quotient Rule.
• Simplify the resulting expression.

In the given exercise, combining \( \log_{10} 30 \) and \( \log_{10} 2 \) involves using the Product Rule because they have the same base of 10.
This simplification leads to a single logarithm \( \log_{10} 60 \). Combining logarithms effectively reduces complexity and makes calculation more straightforward, especially in algebraic or calculus-related problems.