Problem 102 - College Algebra with Modeling and Visualization

Question

Explain why the sequence \(\log 2, \log 4, \log 8, \log 16, \ldots\) is an arithmetic sequence.

Step-by-Step Solution

Answer

The sequence is arithmetic because the difference between terms is consistently \(\log 2\).

Step 1: Understanding the Sequence

The sequence given is \(\log 2, \log 4, \log 8, \log 16, \ldots\). Each term in the sequence appears to be a logarithm of powers of 2. Let's write these in a general form as \(\log (2^n)\) where \(n\) is an integer starting from 1.

Step 2: Expressing Terms Mathematically

The first few terms in the sequence can be rewritten using the power rule of logarithms: \(\log 2 = \log(2^1), \log 4 = \log(2^2), \log 8 = \log(2^3), \log 16 = \log(2^4), \ldots\). In general, the \(n\)-th term is given by \(\log(2^n)\).

Step 3: Applying the Power Rule

Applying the power rule of logarithms, \(\log(a^b) = b \cdot \log a\), we can rewrite \(\log(2^n)\) as \(n \cdot \log 2\). Therefore, the terms in the sequence become \(1 \cdot \log 2, 2 \cdot \log 2, 3 \cdot \log 2, 4 \cdot \log 2, \ldots\).

Step 4: Identifying Common Difference

The sequence \(1 \cdot \log 2, 2 \cdot \log 2, 3 \cdot \log 2, 4 \cdot \log 2, \ldots\) has a constant difference between consecutive terms. This common difference is \(\log 2\) (since each term increases by \(\log 2\)).

Step 5: Conclusion

Since the sequence can be expressed as \(n \cdot \log 2\), and the difference between consecutive terms is constant (\(\log 2\)), it fits the definition of an arithmetic sequence.

Key Concepts

Logarithms

Logarithms are mathematical tools used to deal with the concept of exponentiation. They essentially allow us to transform multiplicative relationships into additive ones, which can simplify complex calculations. A logarithm answers the question: "To what power should we raise a base to get a specific number?" For example, when we write \( \log_2 8 = 3 \), it means raising the base 2 to the power of 3 yields 8.In our sequence example, each term is the logarithm of a power of 2. Using logs, we can express each of these terms succinctly and identify patterns that might not be readily apparent in their original form. The logarithmic properties, such as the power rule, make it easier to manipulate and understand these expressions.

Power Rule

The power rule is a fundamental property of logarithms that states: \( \log(a^b) = b \cdot \log a \). It allows us to bring exponents down as coefficients, making the expression easier to handle and understand.In the sequence \( \log 2, \log 4, \log 8, \log 16, \ldots \), each term is a power of 2. By applying the power rule:

• \( \log(2^1) = 1 \cdot \log 2 \)
• \( \log(2^2) = 2 \cdot \log 2 \)
• \( \log(2^3) = 3 \cdot \log 2 \)
• \( \log(2^4) = 4 \cdot \log 2 \)

This shows that the logarithmic function simplifies complex exponential growth into a simple arithmetic sequence. The power rule is critical in identifying the underlying patterns and making predictions about future terms in the sequence.

Common Difference

In an arithmetic sequence, the same value is consistently added to reach the next term. This value is known as the common difference. Identifying the common difference is key to understanding the sequence type.When we look at the sequence expressions: \( 1 \cdot \log 2, 2 \cdot \log 2, 3 \cdot \log 2, 4 \cdot \log 2, \ldots \), it becomes clear that each term increases by \( \log 2 \). Thus, \( \log 2 \) is the common difference.This consistent increase by \( \log 2 \) in this sequence confirms it as an arithmetic sequence. Recognizing this pattern allows us to predict any term in the sequence effectively. Just by adding or subtracting the common difference, you can move to the next or previous term. This forms the backbone of how arithmetic sequences behave and why they are easy to work with.