Problem 83 - College Algebra

Question

In Exercises \(83-88,\) let \(\log _{b} 2=A\) and \(\log _{b} 3=\) C. Write each expression in terms of \(A\) and \(C\). $$ \log _{b} \frac{3}{2} $$

Step-by-Step Solution

Answer

The expression \(\log _{b} \frac{3}{2}\) in terms of \(A\) and \(C\) is \(C - A\).

Step 1: Substitute values

To begin with, \(\log _{b} \frac{3}{2}\) should be written as a difference of two log values, using the rule \(\log_b \frac{M}{N} = \log_b M - \log_b N\). This gives us \(\log_b 3 - \log_b 2.\)

Step 2: Application of Logarithm values

Next, replace \(\log_b 3\) and \(\log_b 2\) by their values \(C\) and \(A\) respectively. Therefore, \(\log_b 3 - \log_b 2\) becomes \(C - A\).

Step 3: Conclusion

The expression \(\log _{b} \frac{3}{2}\) in terms of \(A\) and \(C\) is \(C - A\).

Key Concepts

Logarithmic Identities

One of the fundamental aspects of understanding logarithms is the use of logarithmic identities. These identities, essentially, are rules that help simplify complicated logarithmic expressions and make calculations easier. They are the backbone of working with logarithms in algebra.The identity used in the problem you encountered is the quotient rule for logarithms. It states:

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

This allows us to express a logarithm of a quotient as a difference between two logarithms. This is particularly useful because it lets us break down complex expressions into simpler parts.When you have values like \( \log_b 3 \) and \( \log_b 2 \), you can utilize these identities to easily plug in known quantities (such as \( C \) and \( A \)) into more complex expressions. This simplification is key to mastering logarithms as it helps you see relationships and results without overwhelming complexity.

Properties of Logarithms

Logarithms follow certain properties that make them quite interesting and versatile in mathematics. These properties not only simplify expressions but also allow logarithms to be effectively used across different mathematical applications.There are a few basic properties that every student should be familiar with:

• Product Rule: \( \log_b(MN) = \log_b M + \log_b N \)
• Quotient Rule: Used in your exercise, \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \)
• Power Rule: \( \log_b(M^p) = p \cdot \log_b M \)

Each of these properties helps break down complex logarithmic expressions into manageable parts, making it easier to solve equations or evaluate expressions. They stem from the basic relationship between exponents and logarithms, giving a reciprocal perspective on powers that is insightful yet practical in real-world applications.

Change of Base Formula

The change of base formula is particularly interesting when it comes to simplifying or evaluating logarithms that aren't in base ten or natural logarithms (base \( e \)). In situations where computational tools only support certain bases, this formula becomes essential.The change of base formula states:

• For any positive numbers \(a, b,\) and \(x\) (where \(b eq 1\)), \( \log_b x = \frac{\log_a x}{\log_a b} \).

This allows you to compute logarithms with an arbitrary base using the logarithms of a new base that might be more convenient or accessible, such as 10 or \( e \). For example, if you need to evaluate \( \log_2 8 \) and only have a calculator for base 10, you could compute it as \( \log_{10} 8 / \log_{10} 2 \). This concept is crucial for enabling comparison and analysis across different systems with ease.