Problem 11 - Algebra A Combined Function

Question

Write each difference as a single logarithm. Assume that variables represent positive numbers. $$ \log _{3} 8-\log _{3} 2 $$

Step-by-Step Solution

Answer

\( \log_3 4 \)

Step 1: State the Logarithmic Difference Rule

According to the properties of logarithms, the difference of two logarithms with the same base can be expressed as a single logarithm by dividing the arguments. In general, \( \log_b a - \log_b c = \log_b \left( \frac{a}{c} \right) \).

Step 2: Apply the Rule to Given Problem

Identify the logarithms from the problem, which are \( \log_3 8 \) and \( \log_3 2 \). Apply the logarithmic difference rule: \[ \log_3 8 - \log_3 2 = \log_3 \left( \frac{8}{2} \right) \].

Step 3: Simplify the Fraction

Simplify the fraction \( \frac{8}{2} \) which is equal to 4. So the expression becomes:\[ \log_3 4 \].

Key Concepts

Logarithmic Difference Rule

The logarithmic difference rule is a powerful property that simplifies expressions involving the difference of two logarithms with the same base. It states that for any positive numbers \(a\) and \(c\), and a base \(b\), the expression \( \log_b a - \log_b c \) can be combined into a single logarithm:

• \( \log_b \left( \frac{a}{c} \right) \)

This rule essentially tells us that subtracting one logarithm from another is equivalent to taking the logarithm of the quotient of their arguments. By leveraging this property, you can transform complex logarithmic expressions into simpler ones, making calculations more straightforward. Understanding this rule is particularly useful when working with expressions in algebra and calculus.
Moreover, it's crucial in simplifying logarithmic equations and solving real-world problems involving exponential growth or decay.

Logarithms with Same Base

Logarithms with the same base are integral to applying the logarithmic difference rule. When two logarithms share the same base, you can easily manipulate them using the property mentioned earlier. It is important to ensure that the base remains consistent throughout the operation to correctly apply the rules.

• For example, in the expression \( \log_3 8 - \log_3 2 \), both logarithms have the base of 3.

Maintaining the same base is important because logarithmic properties, such as the difference rule, are valid only when the bases match. Without this consistency, the simplification or combination of logarithms is not valid.
Always check that the logarithms have the same base before applying any property involving logarithms. This ensures precision and accuracy in your calculations, helping to avoid errors.
Additionally, logarithms with the same base are helpful in solving log equations and understanding exponential relationships in various scientific fields.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions is an essential skill that enables you to handle more complex algebraic problems with ease. Once you've identified that the logarithms share the same base, and applied the appropriate rules, the next step is to simplify the resulting expression.

• For instance, after applying the logarithmic difference rule to \( \log_3 8 - \log_3 2 \), you obtain \( \log_3 \left( \frac{8}{2} \right) \).
• Simplifying the fraction \( \frac{8}{2} \) gives 4, so the expression becomes \( \log_3 4 \).

Reduction to a simpler form such as \(\log_3 4\) not only makes calculations simpler but also makes the expression easier to analyze or evaluate further.
Whether in mathematical exercises, scientific computations, or real-life scenarios involving logarithmic scales, simplification aids in clearer and more efficient problem-solving. By practicing the simplification of logarithmic expressions, you can enhance your understanding and application of logarithmic concepts significantly.