91Ó°ÊÓ

The quotient rule for logarithms is a handy tool in math. It helps in breaking down complex logarithmic expressions. The rule states:

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

This rule is applicable when you have a fraction inside a logarithm. By using the quotient rule, you can split this fraction into separate parts.
For our example, applying the quotient rule allows us to express:

• \( \log_b\left( \frac{x^3 y^2}{z^4} \right) \)
• as \( \log_b(x^3 y^2) - \log_b(z^4) \)

This step simplifies the computation by separating it into two simpler logarithmic expressions. It's a powerful technique to simplify fractions inside logarithms.

The product rule for logarithms deals with multiplying terms inside a logarithm. This rule states:

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

You use this rule when you encounter a product inside a logarithmic expression. The rule helps convert the product into a sum of two logarithms.
In our original problem, for the term:

• \( \log_b(x^3 y^2) \)

we apply the product rule to express it as:

• \( \log_b(x^3) + \log_b(y^2) \)

This simplifies the task by focusing on each factor inside the logarithm separately. Applying the product rule is a crucial step in breaking down the expression further.

The power rule is used when an exponent is involved in the logarithmic expression. According to this rule:

• \( \log_b(M^n) = n \log_b(M) \)

The power rule allows you to take an exponent out of the logarithm, turning it into a multiplier. This can make complex equations much easier to manage.
For example, in our exercise, it was used on terms like:

• \( \log_b(x^3) = 3 \log_b(x) \)

By applying the power rule, each instance of an exponent can be expressed as a product of the exponent and the logarithm. Similarly, for \( \log_b(z^4) \) it becomes:

• \( 4 \log_b(z) \)

This transformation simplifies multiplying terms and allows for easy integration with other rules like the quotient and product rules. It provides a pathway to simplify and evaluate these expressions effectively.