Problem 51 - Precalculus

Question

Evaluate the logarithms using the change-of-base formula. Round to four decimal places. $$\log _{1 / 2} 5$$

Step-by-Step Solution

Answer

\( \log_{1/2} 5 \approx -2.3220 \).

Step 1: Understand the Change-of-Base Formula

The change-of-base formula for logarithms allows us to evaluate logs with any base using common logarithms (base 10) or natural logarithms (base e). The formula is: \[ \log_b a = \frac{\log_c a}{\log_c b} \] where \(c\) is usually 10 or \(e\).

Step 2: Apply the Change-of-Base Formula

We need to find \( \log_{1/2} 5 \). Using the change-of-base formula with common logarithms (base 10), it becomes: \[ \log_{1/2} 5 = \frac{\log_{10} 5}{\log_{10} (1/2)} \]

Step 3: Calculate Each Logarithm Component

Using a calculator, find each logarithm:- \( \log_{10} 5 \approx 0.69897 \)- \( \log_{10} (1/2) \approx -0.30103 \)

Step 4: Divide the Results

Now, divide the results from Step 3: \[ \log_{1/2} 5 = \frac{0.69897}{-0.30103} \approx -2.3220 \]

Step 5: Round to Four Decimal Places

The final value is already calculated to four decimal places, so no further rounding is needed: \[ \log_{1/2} 5 \approx -2.3220 \].

Key Concepts

Understanding Logarithms

Logarithms are the mathematical operations that help us understand how many times we need to multiply a certain number, known as the base, to obtain another number. For example, if we have \( b^x = a \), then \( x \) is the logarithm of \( a \) with base \( b \), written as \( \log_b a \). Logarithms provide a way to simplify complex calculations, especially when dealing with exponential growth or decay.

• Logarithms convert multiplication into addition, which makes them useful in various fields such as science and engineering.
• They are the inverse operations of exponentiation.
• Different bases can serve different purposes; the most common are base \( 10 \) and base \( e \).

By understanding logarithms and their properties, you can better handle calculations involving exponential relationships and simplify your work with a wide range of applications.

Common Logarithm

The common logarithm is a specific type of logarithm with base \( 10 \). It is denoted as \( \log_{10} a \) or simply \( \log a \). Common logs are frequently used in everyday calculations because they simplify working with base \( 10 \), which aligns with our decimal number system.

• For example, \( \log_{10} 100 = 2 \) because \( 10^2 = 100 \).
• Calculators are typically equipped to handle common logarithms, making it easy to compute values.
• The change-of-base formula utilizes common logarithms to calculate any logarithm, regardless of its base.

Knowing about common logarithms can streamline problem-solving involving powers of ten and simplify tasks like scientific notation conversion.

Natural Logarithm

Natural logarithms have a distinct base \( e \), where \( e \) is approximately 2.71828. They are represented as \( \ln a \), and are commonly used in fields that involve continuous growth, such as biology, finance, and certain areas of physics.

• The constant \( e \) plays a critical role in calculus and many mathematical models.
• Natural logs help describe natural growth processes, such as population growth or radioactive decay.
• They provide a foundation for understanding concepts like compounding interest and change rates.

When you familiarize yourself with natural logarithms, you'll find they are indispensable in understanding and analyzing growth and decay processes in natural and theoretical contexts.