Problem 92 - A Graphical Approach to Precalculus with Limits

Question

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{1 / 2} 3$$

Step-by-Step Solution

Answer

\( \log_{1/2} 3 \approx -1.585 \).

Step 1: Understand the Change-of-Base Formula

The change-of-base formula allows us to rewrite a logarithm in terms of logs with different bases. The formula is: \[ \log_b a = \frac{\log_c a}{\log_c b} \]where \( c \) can be any positive number (except 1). A common choice of \( c \) is 10 or \( e \) because we have calculators that directly compute \( \log_{10} \) (common logarithm) or \( \ln \) (natural logarithm).

Step 2: Apply the Change-of-Base Formula

For the logarithm \( \log_{1/2} 3 \), apply the change-of-base formula to express it in terms of base 10 (common logarithms):\[ \log_{1/2} 3 = \frac{\log_{10} 3}{\log_{10} (1/2)} \].

Step 3: Calculate \( \log_{10} 3 \) and \( \log_{10} (1/2) \)

Using a calculator, evaluate the following:\( \log_{10} 3 \approx 0.4771 \)\( \log_{10} (1/2) = \log_{10} 1 - \log_{10} 2 \approx 0 - 0.3010 = -0.3010 \).

Step 4: Compute the Approximation

Plug the values obtained from the previous step into the change-of-base expression:\[ \log_{1/2} 3 = \frac{0.4771}{-0.3010} \approx -1.585 \].

Step 5: Interpret the Result

The result \( -1.585 \) indicates the power to which the base \( \frac{1}{2} \) must be raised to obtain the number 3.

Key Concepts

Logarithms

Logarithms are mathematical operations that help us understand exponential relationships. They tell us how many times we must multiply a base number to obtain another number. For example, in the expression \( \log_2 8 \), we ask: "How many times must we multiply 2 to get 8?" The answer is 3, because \( 2^3 = 8 \). The logarithm is a tool to reverse the process of exponentiation. It simplifies complex calculations by converting multiplication and division into addition and subtraction.

• Notation: \( \log_b a \) means logarithm of \( a \) with base \( b \)
• Inverse Relation: \( b^{\log_b a} = a \)
• Common Bases: 10 (common logarithm) and \( e \) (natural logarithm)

Understanding logarithms is crucial in many fields, including science, engineering, and finance. They can describe phenomena such as loudness, pH levels, and earthquake magnitudes.

Base Conversion

Base conversion is the process of changing the base of a logarithm to make calculations easier. By converting the base, we can use logs that are more familiar or accessible, such as common logs or natural logs.
The change-of-base formula is a key tool in this process, given by:\[\log_b a = \frac{\log_c a}{\log_c b}\]Here, \( c \) can be any positive number, but common choices are 10 and \( e \), since calculators often have functions for these bases.

• Flexibility: Allows choosing a convenient base for easier computation
• Common Bases: Base 10 (\( \log_{10} \)) and Base \( e \) (\( \ln \)) are widely used
• Simplification: Using the formula can simplify solving logs with uncommon bases

Converting the base through this method makes it possible to solve logarithms on devices and calculators that only support certain bases, broadening the practical applications of logarithmic calculations.

Common Logarithm

A common logarithm is a logarithm with base 10, often referred to simply as \( \log \) without any subscript. This base is prevalent because it aligns with our typical number system, which also uses base 10. Many calculators have a \( \log \) button that computes logarithms of this type directly.
Using common logarithms simplifies calculations by making use of built-in calculator functions. Most scientific calculators can compute these directly, providing a quick way to find log values for numbers.

• Notation: Written as \( \log_{10} \) or just \( \log \)
• Calculators: Devices are often equipped with a \( \log \) button
• Utility: Useful in diverse applications, from engineering to finance

While representing any log, it's essential to confirm the base as 10, especially in contexts where multiple bases are used. Understanding common logarithms is foundational, as they are utilized often in academic and professional settings to streamline calculations and interpretations.