Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

How are \(\log _{1 / 2} x\) and \(\log _{2} x\) related?

Step-by-Step Solution

Answer

\(\log_{1/2} x = -\log_2 x\). They are negatives of each other.

Step 1: Understanding the Logarithm Change of Base Formula

To relate the two logarithms given, we can use the change of base formula for logarithms. The formula is: \[\log_b a = \frac{\log_k a}{\log_k b}\]This formula allows us to convert the base of any logarithm into another base, making calculations and comparisons between logarithms easier.

Step 2: Converting \\(ackslash\log _{1/2} x\\) using the change of base formula

Let's first express \(\log_{1/2} x\) with base 2 using the change of base formula. Set \(k = 2\):\[\log_{1/2} x = \frac{\log_2 x}{\log_2 (1/2)}\]

Step 3: Calculate \\(ackslash\log_2 (1/2)\\)

To find \(\log_2 (1/2)\), we use the property of logarithms: \(\log_b b^c = c\). The mathematical expression \(1/2 = 2^{-1}\), so:\[\log_2 (1/2) = \log_2 (2^{-1}) = -1\]

Step 4: Substituting values and simplifying

Substitute \(\log_2 (1/2) = -1\) into the expression for \(\log_{1/2} x\):\[\log_{1/2} x = \frac{\log_2 x}{-1} = -\log_2 x\]

Step 5: Conclusion

The relationship between \(\log_{1/2} x\) and \(\log_2 x\) is that \(\log_{1/2} x = -\log_2 x\). Therefore, they are negatives of each other, meaning as one increases, the other decreases.

Key Concepts

Change of Base Formula

The change of base formula is a fundamental tool when dealing with logarithms of different bases. This formula comes in handy because it allows us to recalibrate any logarithm to a more feasible base, usually base 10 or base 2, which are readily accessible on calculators. The formula itself is given as:

• \[ \log_{b} a = \frac{\log_{k} a}{\log_{k} b} \]

This formula operates by expressing one logarithm as a fraction of two others, offering a seamless way to convert bases. By choosing an appropriate common base \( k \), such as base ten (\( \log_{10} \)) or base two (\( \log_{2} \)), calculations become much more manageable.
When employing the change of base formula, it's essential first to understand the bases you are converting between. While it's often used to switch from a less common or complex base to a more familiar one, it also helps discover relationships between logarithms, as is the task at hand in this exercise.

Logarithmic Properties

Logarithmic properties serve as core pillars in understanding how logarithms operate and interact with each other. These properties simplify complex expressions and uncover hidden relationships between logarithms of different bases. Here are some key properties:

• Product Property: \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \) This property states the logarithm of a product is the sum of the logarithms.
• Quotient Property: \( \log_{b}\left(\frac{m}{n}\right) = \log_{b}(m) - \log_{b}(n) \) The logarithm of a quotient is the difference of the logarithms.
• Power Property: \( \log_{b}(a^c) = c \cdot \log_{b}(a) \) This emphasizes that exponents can be brought down as coefficients.
• Change of Base Property: \( \log_{b} a = \frac{\log_{k} a}{\log_{k} b} \) Another example of changing one logarithm base to another.

In this particular exercise, the conversion of \( \log_{2}(1/2) \) relies on recognizing that \( 1/2 \) is the same as \( 2^{-1} \). Applying the Power Property, we find \( \log_{2}(1/2) = -1 \). This realization demonstrates the elegance of logarithmic properties in revealing the negation involved when comparing \( \log_{1/2} x \) and \( \log_2 x \).

Base Conversion

Base conversion in logarithms specifically refers to the act of translating a given logarithm from one base to another, often to simplify the calculation or comparison process. When tackling problems like deriving the relationship between \( \log_{1/2} x \) and \( \log_{2} x \), understanding base conversion is essential.
The method takes advantage of the change of base formula, which facilitates viewing different logarithm bases through a shared lens. This allows us to determine equivalences or transforms between logarithms efficiently. Let's see how it works in the context of our exercise:

• Start with \( \log_{1/2} x \).
• Use the change of base formula: \[ \log_{1/2} x = \frac{\log_{2} x}{\log_{2} (1/2)} \].
• Recognize \( \log_{2} (1/2) = -1 \) due to the expression \( 1/2 = 2^{-1} \).
• Substitute in and simplify: \( \log_{1/2} x = -\log_2 x \).

This conversion reveals that \( \log_{1/2} x \) is the negation of \( \log_{2} x \), fundamentally altering our perspective on such relationships. These steps underscore the significant role of base conversion in opening new pathways of mathematical insight.