Problem 24 - Algebra and Trigonometry

Question

Use the Laws of Logarithms to expand the expression. $$ \log _{a}\left(\frac{x^{2}}{y z^{3}}\right) $$

Step-by-Step Solution

Answer

The expanded form is \( 2\log_a(x) - \log_a(y) - 3\log_a(z) \).

Step 1: Apply the Quotient Rule

The quotient rule for logarithms states that \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \). Apply this rule to the given expression: \( \log_a \left( \frac{x^2}{yz^3} \right) = \log_a(x^2) - \log_a(yz^3) \).

Step 2: Expand using the Product Rule

The product rule for logarithms states that \( \log_b(M \times N) = \log_b(M) + \log_b(N) \). Apply this rule to the second term: \( \log_a(yz^3) = \log_a(y) + \log_a(z^3) \). Substitute back into the expression: \( \log_a(x^2) - (\log_a(y) + \log_a(z^3)) \).

Step 3: Apply the Power Rule

The power rule for logarithms states that \( \log_b(M^n) = n\log_b(M) \). Apply this rule to the terms: \( \log_a(x^2) = 2\log_a(x) \) and \( \log_a(z^3) = 3\log_a(z) \). Substitute these back: \( 2\log_a(x) - (\log_a(y) + 3\log_a(z)) \).

Step 4: Simplify the Expression

Combine the terms by distributing the negative sign: \( 2\log_a(x) - \log_a(y) - 3\log_a(z) \).

Key Concepts

Quotient Rule

Understanding the quotient rule for logarithms is essential when working with expressions that involve division. Simply put, if you have the logarithm of a quotient, like \( \log_b\left(\frac{M}{N}\right) \), this can be rewritten and simplified by separating it into two distinct logarithms: \( \log_b(M) \) and \( \log_b(N) \).
This transformation uses subtraction, as the logarithm of a division is equal to subtracting the logarithm of the denominator from the numerator's logarithm.
For example, with the expression \( \log_a\left(\frac{x^2}{yz^3}\right) \), applying the quotient rule results in \( \log_a(x^2) - \log_a(yz^3) \). This simplification forms the foundation for further expansions using other logarithmic rules.

Product Rule

The product rule comes into play when dealing with the logarithm of a product, specifically when your expression involves multiplying numbers or variables together. According to this rule, \( \log_b(M \times N) \) can be expanded into \( \log_b(M) + \log_b(N) \).
So, when you encounter something like \( \log_a(yz^3) \), it can be expanded using the product rule to \( \log_a(y) + \log_a(z^3) \).
This step allows you to express products within logarithms as a sum, making it easier to apply additional transformations, like the power rule, to each separate term.

Power Rule

The power rule simplifies the process of dealing with logarithms raised to a power by transforming it into a product. Specifically, \( \log_b(M^n) \) translates to \( n \log_b(M) \).
For instance, in the expression \( \log_a(x^2) \), applying the power rule gives \( 2\log_a(x) \). Similarly, \( \log_a(z^3) \) simplifies to \( 3\log_a(z) \).
This rule is particularly handy as it reduces complex expressions with exponents into straightforward linear expressions, facilitating easier manipulation and further simplification of the original logarithmic expression.