Problem 41 - Algebra A Combined Function

Question

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{4} \frac{2}{9 z} $$

Step-by-Step Solution

Answer

\( \log_{4}(2) - \log_{4}(9) - \log_{4}(z) \)

Step 1: Apply the Quotient Rule of Logarithms

The Quotient Rule of Logarithms states that \( \log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \). Apply this rule to the expression \( \log_{4} \left( \frac{2}{9z} \right) \). This gives us: \[ \log_{4} \left( \frac{2}{9z} \right) = \log_{4}(2) - \log_{4}(9z) \].

Step 2: Apply the Product Rule to the Denominator

The expression \( \log_{4}(9z) \) in the denominator is a product. The Product Rule of Logarithms states \( \log_b(MN) = \log_b(M) + \log_b(N) \). Apply this rule to get: \[ \log_{4}(9z) = \log_{4}(9) + \log_{4}(z) \].

Step 3: Combine the Logarithmic Expressions

Substitute back the expression from Step 2 into the equation from Step 1: \[ \log_{4}(2) - (\log_{4}(9) + \log_{4}(z)) \]. Simplify using the distributive property of subtraction to get the final expression: \[ \log_{4}(2) - \log_{4}(9) - \log_{4}(z) \].

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms helps in breaking down complex logarithms that involve division into simpler terms. When you have an expression like \( \log_b \left( \frac{M}{N} \right) \), according to the quotient rule, it can be expressed as \( \log_b(M) - \log_b(N) \). This translates the division inside the logarithm into a subtraction of two separate logarithms.
For example, if you have \( \log_4 \left( \frac{2}{9z} \right) \), applying the quotient rule simplifies this to \( \log_4(2) - \log_4(9z) \). This makes solving and simplifying logarithmic expressions much more manageable and easier to deal with.
This rule is particularly handy when simplifying expressions where you need to write them as a sum or difference of logs. Understanding this property helps in systematically reducing complex expressions into simpler ones, which are often easier to interpret and manage.

Product Rule of Logarithms

Just as the quotient rule helps with division, the product rule of logarithms aids in simplifying logarithms involving multiplication. The rule states that \( \log_b(MN) = \log_b(M) + \log_b(N) \). Here, the multiplication turns into addition, allowing us to write the expression as a sum of individual logarithms.
In the given example, after using the quotient rule, we encountered \( \log_4(9z) \) in the denominator. By applying the product rule, this expression can be expanded to \( \log_4(9) + \log_4(z) \). This is crucial in breaking down more complex logarithms into simpler, more digestible parts.
Using the product rule in combination with other logarithmic properties enhances your ability to transform and simplify logarithmic expressions effectively, ensuring a thorough understanding of their underlying components.

Properties of Logarithms

Logarithms have numerous properties that are instrumental in converting complex expressions into simpler, more understandable forms. Besides the quotient and product rules mentioned earlier, another key property is the power rule, which states \( \log_b(M^n) = n \cdot \log_b(M) \). This allows you to bring down powers within a logarithm, simplifying expressions even further.
These properties help in the systematic manipulation of logarithmic expressions. Take the original problem again: after applying the quotient and product rules, we arrive at \( \log_4(2) - \log_4(9) - \log_4(z) \). By using the various logarithmic properties, you can clearly see how to deconstruct and simplify previously intimidating expressions.
Understanding these principles not only aids in homework problems but ensures a broader competence in dealing with mathematical problems involving logarithms, preparing students better for advanced mathematical topics.