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The quotient rule in logarithms is an essential property that helps to simplify expressions involving the division of numbers or variables. According to the quotient rule, when you have the logarithm of a fraction, such as \( \log_a \frac{b}{c} \), it can be rewritten as the difference of two separate logarithms: \( \log_a b - \log_a c \).
This rule is particularly helpful because it breaks down complex expressions into simpler components.

For example, in the expression \( \log_2 \frac{2 \sqrt{3}}{5p} \), the quotient rule allows us to express it as \( \log_2 (2 \sqrt{3}) - \log_2 (5p) \).
This transformation turns one challenging fraction into two straightforward terms, setting the stage for further simplification.

The product rule of logarithms is another important tool for simplifying logarithmic expressions. This rule tells us that the logarithm of a product can be written as the sum of the logarithms of its factors. Mathematically, it appears as \( \log_a (b \, c) = \log_a b + \log_a c \).
The product rule essentially transforms a single compounded term into two simpler parts, allowing for easier manipulation and understanding.

In this exercise, the expression \( \log_2 (2 \sqrt{3}) \) can be decomposed using the product rule into \( \log_2 2 + \log_2 \sqrt{3} \).
This breakdown simplifies the logarithmic process, making each term easier to work with individually.

The power rule of logarithms provides a way to deal with expressions where numbers or variables are raised to a power.
This rule states that \( \log_a (b^c) = c \cdot \log_a b \), allowing you to "bring down" the exponent as a coefficient in front of the logarithm.

Consider the portion of our exercise involving \( \log_2 \sqrt{3} \). Knowing that \( \sqrt{3} \) is equivalent to \( 3^{1/2} \), this can be rewritten using the power rule: \( \log_2 (3^{1/2}) = \frac{1}{2} \log_2 3 \).
By turning the exponent into a multiplier, it becomes easier to manage logarithmic expressions with exponents.

Simplifying logarithmic expressions often involves the application of the quotient, product, and power rules, as seen in the provided problem.
To reach a fully simplified outcome, each term is methodically broken down to its simplest components and then combined back into a coherent expression.

In our specific exercise, after applying all rules and simplifying known values such as \( \log_2 2 = 1 \), the expression was gradually transformed.

• First, the application of rules broke down the expression \( \log_2 \frac{2 \sqrt{3}}{5p} \).
• Next, known values and rules condensed it into \( 1 + \frac{1}{2} \log_2 3 - \log_2 5 - \log_2 p \).

This final form effectively expresses the initial fraction in a simplified manner, illustrating the power and utility of logarithmic properties.