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When working with logarithms, simplifying an expression down to a single logarithm can make computations much easier. In the context of the given problem, rewriting the expression as a single logarithm involves combining separate logarithmic terms into one. This process allows us to leverage various properties and rules associated with logarithms, making expressions easier to handle and simplifying calculations.
When an expression consists of the subtraction or addition of two or more logarithms, it can often be reduced to a single logarithmic expression using logarithm rules. This simplifies not only the expression itself but also any further computations or evaluations that need to be performed.

Logarithm rules are integral in converting and simplifying logarithmic expressions. The key logarithm rules include:

• Power Rule: This states that a logarithm of a number multiplied by a coefficient can be rewritten as a logarithm of the number raised to that power. For example, \( k \log_b a = \log_b a^k \).
• Product Rule: It says that the logarithm of a product is the sum of the logarithms of the factors, i.e., \( \log_b(XY) = \log_b X + \log_b Y \).
• Quotient Rule: This states that the logarithm of a quotient is the difference of the logarithms, i.e., \( \log_b\left(\frac{X}{Y}\right) = \log_b X - \log_b Y \).

In this exercise, the Power Rule was used to first rewrite \( 2 \ln 3 \) as \( \ln 3^2 \). Then, by applying the Quotient Rule, the expression \( \ln 9 - \ln 9 \) simplifies to \( \ln\left(\frac{9}{9}\right) \). Understanding and utilizing these rules are crucial for simplifying complex logarithmic expressions effectively.

Making complex logarithmic expressions simpler often involves applying the appropriate logarithm rules. In this problem, we begin with \( \ln 9 - 2 \ln 3 \).

• First, notice the coefficient '2' in front of \( \ln 3 \). Using the Power Rule, this becomes \( \ln 3^2 \), which is \( \ln 9 \).
• Next, look at the subtraction \( \ln 9 - \ln 9 \). The Quotient Rule allows us to rewrite this as a single logarithm: \( \ln\left(\frac{9}{9}\right) \).
• Finally, evaluate \( \ln\left(\frac{9}{9}\right) = \ln 1 \), which simplifies to 0, because the natural logarithm of 1 is always 0.

This logical sequence of simplification highlights the efficiency of understanding logarithm properties and applying them methodically. Recognizing when and how to use these rules simplifies the process and enhances your computational prowess in dealing with logarithms.