91Ó°ÊÓ

The quotient rule of logarithms is a fundamental property used to handle the logarithm of a division of two quantities. This property can be extremely useful for simplifying complex logarithmic expressions. It states that for any positive numbers \( M \) and \( N \), and a positive base \( b \), we have:

• \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \)

This rule allows us to separate the division inside a logarithm into the difference between two individual logarithms. For example, let's say we have \( \log_6 \left( \frac{y^5 w^2}{x^4 z^3} \right) \). According to the quotient rule, this can be rewritten as:

• \( \log_6 (y^5 w^2) - \log_6 (x^4 z^3) \)

This transformation simplifies calculations and helps clarify the relationships between different parts of the original logarithmic expression.

The product rule of logarithms is another important property which simplifies the logarithm of a product of two or more numbers. The rule states that the logarithm of a product is the sum of the logarithms of its factors. For positive numbers \( M \) and \( N \), and a positive base \( b \), it can be expressed as:

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

Using this rule, we break down complex expressions easily. For instance, after applying the quotient rule, one of our expressions might be \( \log_6 (y^5 w^2) \). By using the product rule, this becomes:

• \( \log_6 (y^5) + \log_6 (w^2) \)

Similarly, \( \log_6 (x^4 z^3) \) becomes:

• \( \log_6 (x^4) + \log_6 (z^3) \)

These adjustments make further manipulations, like applying the power rule, much more straightforward.

The power rule of logarithms is vital when dealing with logarithmic expressions involving exponents. It offers a way to "take out" an exponent present in a logarithmic expression, making the expression simpler to manage. The rule states that for any positive number \( M \), an exponent \( n \), and a positive base \( b \), the following holds:

• \( \log_b (M^n) = n \log_b M \)

By employing the power rule, we can simplify expressions containing powers within logarithms. Consider the expression from our problem statement: after applying the quotient and product rules, we arrive at terms like \( \log_6 (y^5) \) and \( \log_6 (w^2) \). By using the power rule, these become:

• \( 5 \log_6 y \)
• \( 2 \log_6 w \)

Similarly, for terms \( \log_6 (x^4) \) and \( \log_6 (z^3) \), the expressions simplify to:

• \( 4 \log_6 x \)
• \( 3 \log_6 z \)

These transformations allow for the final expression to be simplified into a straightforward combination of individual logarithms for \( x, y, z, \) and \( w \).