Problem 6 - College Algebra with Corequisite Support

Question

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\log _{4}\left(\frac{\frac{x}{z}}{w}\right)\)

Step-by-Step Solution

Answer

\( \log_4(x) - \log_4(z) - \log_4(w) \)

Step 1: Understand the Expression

The expression given is \( \log _{4}\left(\frac{\frac{x}{z}}{w}\right) \). We need to break it down using logarithmic properties.

Step 2: Apply the Quotient Rule of Logarithms

The quotient rule states that \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \). Apply this rule: \[ \log_4\left(\frac{\frac{x}{z}}{w}\right) = \log_4\left(\frac{x}{z}\right) - \log_4(w) \]

Step 3: Simplify the First Logarithmic Term

We will now use the quotient rule on \( \log_4\left(\frac{x}{z}\right) \). Applying the rule gives:\[ \log_4\left(\frac{x}{z}\right) = \log_4(x) - \log_4(z) \]

Step 4: Combine All Terms

Substitute the result from Step 3 back into the expression from Step 2:\[ \log_4\left(\frac{x}{z}\right) - \log_4(w) \] becomes \[ \log_4(x) - \log_4(z) - \log_4(w) \]

Step 5: Final Expression

The expression expanded as a sum and difference of logs is:\[ \log_4(x) - \log_4(z) - \log_4(w) \]

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is a fundamental concept that helps simplify logarithmic expressions involving division. It allows us to break down complex logarithmic functions into simpler parts by expressing a logarithm of a quotient as a difference of logarithms. The general formula for the quotient rule is:

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

This rule is particularly useful when you encounter a logarithmic expression like \( \log_4\left(\frac{x}{z}\right) \), where the numerator and denominator are both variable expressions. By using the quotient rule, you can split this expression into \( \log_4(x) - \log_4(z) \).
For our original problem, we used this rule multiple times. Initially, it applied to \( \log_4\left(\frac{\frac{x}{z}}{w}\right) \), breaking it down into \( \log_4\left(\frac{x}{z}\right) \) and \(- \log_4(w)\). Then, by applying the quotient rule again on \( \log_4\left(\frac{x}{z}\right) \), we further expanded it into \( \log_4(x) - \log_4(z) \).
Understanding the quotient rule and how to apply it helps in simplifying and solving many logarithmic problems, especially those involving division.

Expanding Logarithms

Expanding logarithms is a process of rewriting a single logarithmic expression into multiple terms that are either added or subtracted. This is achieved using properties like the product rule, quotient rule, and power rule. When expanding logarithmic expressions, you dissect them into more manageable parts. This not only makes complex expressions simpler but also aids in solving equations that include logs.
In our example, we successfully expanded \( \log_4\left(\frac{\frac{x}{z}}{w}\right) \) into individual log terms by repeatedly applying the quotient rule. Each time, we simplified the expression further, revealing individual log components. Starting with \( \log_4\left(\frac{x}{z}\right) - \log_4(w) \), using the rule again, we got \( \log_4(x) - \log_4(z) - \log_4(w) \).
The expansion of logarithms is crucial for expressing equations in simpler forms. It makes the recognition of patterns easier and paves the way for evaluating or interpreting logarithmic functions.

Logarithmic Expressions

Logarithmic expressions involve the use of logarithms to describe mathematical relationships, usually involving complex multiplication, division, or powers. Such expressions often require simplification or expansion using properties of logarithms for ease of interpretation or calculation.
In most cases, to simplify or expand these expressions, the following properties are employed:

• Product Rule: \( \log_b(M \cdot N) = \log_b(M) + \log_b(N) \)
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)
• Power Rule: \( \log_b(M^n) = n \cdot \log_b(M) \)

Logarithmic expressions can vary greatly in complexity, from simple equations, as demonstrated with \( \log_4(x) - \log_4(z) - \log_4(w) \), to more intricate ones involving different operations or even logarithms with different bases.
Mastering how to manipulate these expressions using the rules mentioned is key to tackling various mathematical problems, whether algebraic, geometric, or in real-world applications like data science or finance. Thus, understanding the structure and rules surrounding logarithmic expressions is vital for any math enthusiast.