Problem 92 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

\( \log_{1/2} 3 \approx -1.5850 \).

Step 1: Recall the Change of Base Formula

To find an approximation for a logarithm with a non-standard base, we use the change of base formula. The change of base formula for a logarithm \( \log_b a \) is given by \( \frac{\log_c a}{\log_c b} \). Typically, we use base 10 or base \( e \) (natural logarithm) for calculations.

Step 2: Apply the Change of Base Formula

For \( \log_{1/2} 3 \), we apply the formula as follows: \( \log_{1/2} 3 = \frac{\log_{10} 3}{\log_{10} (1/2)} \). You can use base 10 or the natural logarithm (base \( e \)); here, we will use 10 for simplicity.

Step 3: Calculate Each Logarithm

Using a calculator, find \( \log_{10} 3 \) and \( \log_{10} (1/2) \). Approximate these values to several decimal places for accuracy. \( \log_{10} 3 \approx 0.4771 \) and \( \log_{10} (1/2) \approx -0.3010 \).

Step 4: Divide to Find the Approximation

Now divide the values obtained: \( \frac{0.4771}{-0.3010} \approx -1.5850 \). This gives us the approximate value of \( \log_{1/2} 3 \).

Key Concepts

Change of Base Formula

When dealing with logarithms with non-standard bases, the change of base formula is incredibly helpful. It allows us to compute logarithms that aren't in base 10 or base \( e \). The formula states that for a given logarithm \( \log_b a \), you can express it as \( \frac{\log_c a}{\log_c b} \). This means you divide the logarithm of the desired number \( a \) by the logarithm of the base \( b \), both in the same new base \( c \).
For practical calculations, such as using a calculator, we often choose \( c = 10 \) or \( c = e \). These are the most common because calculators are typically designed to handle these bases efficiently. This process does not alter the result of the logarithm; it just makes it possible to calculate using standard tools.

College Algebra

College Algebra introduces students to a wide array of mathematical concepts, including functions, systems of equations, and importantly, logarithms. Logarithms are a way to represent numbers based on exponentials. They transform multiplicative processes into additive ones, which simplify the arithmetic.

• Understanding Logarithms: The concept of a logarithm revolves around finding the power to which a base must be raised to produce a given number.
• Educational Importance: Mastery of logarithms in college algebra paves the way to understanding more complex topics in mathematics and science, such as exponential growth in calculus or natural logs in chemistry.

College Algebra courses emphasize the use of logarithms and their properties, including the change of base formula, to equip students with skills necessary for higher-level math and diverse scientific fields.

Non-Standard Base

A non-standard base in logarithms is any base that isn't commonly used in traditional arithmetic (such as 10, or \( e \)). The base of \( \frac{1}{2} \) in \( \log_{1/2} 3 \) is an example of this, where the base isn't the usual 10 or \( e \).
Understanding base changes is crucial because:

• Flexibility in Problem-Solving: It helps in determining logarithmic values for a variety of bases, which is often required in real-world applications.
• Appropriate Usage: Specific fields or problems might demand calculations in non-standard bases, making this knowledge vital.

By applying the change of base formula, students learn to adapt and solve complex problems that involve non-standard bases without being restricted to familiar ones. This skill is essential in various scientific and engineering disciplines where different bases may arise.