Problem 67 - College Algebra and Calculus: An Applied Approach

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{3} 4 n\)

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression, \(\log _{3} 4n\), is \( \ln(4) / \ln(3) + \log_{3}(n) \).

Step 1: Apply the Product Rule

The Product Rule for logarithms states that for any positive variables a, b, and c, \( \log _{a}(bc) = \log _{a}(b) + \log _{a}(c) \). Apply the Product Rule to the given expression \(\log _{3} 4n\), separate as a sum of two logs: \(\log_{3}(4) + \log_{3}(n)\).

Step 2: Change the Base

The Change of Base Rule for logarithms states that for any positive variables a, b, and c (where a≠1), \( \log _{b}(a) = \log _{c}(a) / \log _{c}(b) \). The logarithm base 3 can be expressed as the ratio of natural logarithms, adjusting the first term in the sum using this rule we get \( \log_{3}(4) = \ln(4) / \ln(3)\).

Step 3: Compose the Final Answer

Combine the results from step 1 and 2. The expanded form of the given logarithm expression is \( \ln(4) / \ln(3) + \log_{3}(n) \).

Key Concepts

Product Rule

The Product Rule is a useful property of logarithms that allows you to break down complex expressions. It states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. For example, given any positive variables like a, b, and c, the rule is expressed as:

• \( \log_{a}(bc) = \log_{a}(b) + \log_{a}(c) \)

This property is fundamental when you encounter expressions that involve multiplication under a logarithm. Let's take the specific case from the exercise, \( \log_{3} 4n \). Here, '4n' is treated as '4' multiplied by 'n'.
By applying the Product Rule, this expression separates into:

• \( \log_{3}(4) + \log_{3}(n) \)

This expansion helps simplify and makes further manipulation easier, especially when dealing with more complex logarithmic expressions.

Change of Base Rule

The Change of Base Rule is another essential tool for managing logarithms, especially when technology or a standard base (like base 10 or e) is required. The rule allows changing the base of a logarithm to a more convenient one. For any positive numbers a, b, and c, where a is not equal to 1, the rule is:

• \( \log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)} \)

This implies you can rewrite logs in terms of other bases, such as common logarithms (base 10) or natural logarithms (base e). In the solved exercise, \( \log_{3}(4) \) can be converted using natural logarithms:

• \( \log_{3}(4) = \frac{\ln(4)}{\ln(3)} \)

This transformation is powerful in situations where the calculation needs to be handled by a calculator or precise value is desired. It also paves the way for simplification and comparison across different logarithmic expressions.

Expansion of Logarithmic Expressions

Expanding logarithmic expressions involves breaking down complex logs into simpler parts using properties like the Product Rule, Quotient Rule, or Power Rule. This is particularly important in algebra and calculus, where simplifying expressions is key.By expanding a logarithmic expression, you can express it as a sum, difference, or a multiple of individual logarithms. The exercise initially given can illustrate this. Starting with \( \log_{3}(4n) \), we use the Product Rule to distinguish each part:

• \( \log_{3}(4) + \log_{3}(n) \)

But the expansion doesn't stop there. Using the Change of Base Rule, \( \log_{3}(4) \) becomes:

• \( \frac{\ln(4)}{\ln(3)} \)

Thus, the complete expanded form is \( \frac{\ln(4)}{\ln(3)} + \log_{3}(n) \).
This process of expansion simplifies the expression, making it easier to evaluate, compare, or manipulate in further mathematical operations.