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The logarithm addition rule is a fundamental property of logarithms that helps in simplifying logarithmic expressions. This rule states that if you have two logarithms with the same base and you are adding them, you can combine them into a single logarithm. The combined logarithm will have its argument as the product of the individual arguments.

Mathematically, this is represented as:
\[ \log_b(m) + \log_b(n) = \log_b(m \times n) \].
For example, if you have \( \log_2(3) + \log_2(4) \), you can combine these into \( \log_2(3 \times 4) = \log_2(12) \). This simplification makes it easier to work with logarithmic expressions and solve equations involving logarithms.

Logarithmic expressions are mathematical expressions that involve logarithms. A logarithm is the inverse operation to exponentiation, meaning it helps to determine what exponent produced a specific number.
In general, if \( b^y = x \), then \( \log_b(x) = y \). In a logarithmic expression, you might encounter various operations, such as addition, subtraction, and multiplication, similar to what we see in basic arithmetic with regular numbers.

When you work with logarithmic expressions, always check the base of the logarithms you're dealing with. To apply simplification rules like the logarithm addition rule, all logarithms in the expression must have the same base.
For instance, in the expression \( \log_a(5) + \log_a(14) \), both logarithms have base \( a \), which allows us to simplify using the logarithm addition rule.

Simplifying logarithms involves using various rules and properties to make a logarithmic expression more manageable or to solve a problem.
One key rule for simplification is the logarithm addition rule, which we used in the initial example exercise. Here's how you apply it: When you have an expression like \( \log_a(5) + \log_a(14) \), you can combine these into a single logarithm: \( \log_a(5 \times 14) \).

You then calculate the product in the argument: \( 5 \times 14 = 70 \), so the final simplified expression becomes \( \log_a(70) \).
By transforming multiple logarithms into one, you make the expression simpler and easier to handle. Other important tools for simplifying logarithms include the logarithm subtraction rule, the power rule, and the change of base formula.

Always remember to follow these steps:

• Check if the bases are the same.
• Apply the appropriate logarithm rule.
• Simplify any arithmetic operations within the argument.

Practicing these steps will help you become more skilled at working with logarithmic expressions and simplifications.