Problem 94 - A Graphical Approach to College Algebra

Question

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{200} 175$$

Step-by-Step Solution

Answer

\( \log_{200} 175 \approx 0.974 \)

Step 1: Understand the Change-of-Base Formula

The change-of-base formula allows us to rewrite a logarithm \(\log_b a\) using logarithms of any other base, commonly base 10 or base e. The formula is: \\[\log_b a = \frac{\log_c a}{\log_c b}\] \where \(c\) is the new base. For this example, we'll use base 10.

Step 2: Apply the Change-of-Base Formula

We will rewrite \(\log_{200} 175\) using base 10: \\[\log_{200} 175 = \frac{\log_{10} 175}{\log_{10} 200}\] \This expression allows us to compute the logarithm using a calculator, which typically supports log base 10.

Step 3: Calculate Each Logarithm

Use a calculator to compute \(\log_{10} 175\) and \(\log_{10} 200\): \- \(\log_{10} 175 \approx 2.2430\) \- \(\log_{10} 200 \approx 2.3010\)

Step 4: Divide to Find the Approximation

Now divide the two results from Step 3 using the formula from Step 2: \\[\frac{\log_{10} 175}{\log_{10} 200} \approx \frac{2.2430}{2.3010} \approx 0.974\] \Thus, \(\log_{200} 175 \approx 0.974\).

Key Concepts

Logarithms: Understanding the Basics

Logarithms are a fundamental concept in mathematics, especially in algebra and calculus. They help to simplify expressions involving exponents by turning them into manageable numbers. When you see a logarithm written as \( \log_b a \), it asks the question: "What power do we need to raise \( b \) to, in order to get \( a \)?"
For example, in \( \log_2 8 \), we are asking what power do we need to raise 2 to, in order to get 8. The answer is 3, because \( 2^3 = 8 \). This makes logarithms extremely useful for solving equations where the unknown is an exponent. They are built specifically to unravel those tricky exponent numbers.
In practical scenarios:

• Logarithms are used in science, engineering, and finance to deal with data that span large ranges.
• They help in simplifying calculations, especially those involving exponential growth and decay.

By understanding logarithms, you can open the door to easier calculations and deeper insights into mathematical relationships.

Base Conversion: Making Calculations Easier

Base conversion in logarithms allows us to work with different bases easily. The core idea is to translate a logarithm in one base to another base, more commonly base 10, which is widely supported by calculators. This is where the change-of-base formula comes into play.
The formula is written as:\[ \log_b a = \frac{\log_c a}{\log_c b} \]where \( c \) is the new base, generally 10 or \( e \) (natural logarithm). This transformation simplifies calculations significantly.
For example, to transform \( \log_{200} 175 \) using base 10, we express it as:\[ \log_{200} 175 = \frac{\log_{10} 175}{\log_{10} 200} \]This re-expression allows you to easily calculate the logarithm using standard scientific calculators.
Why change base?

• It converts complex logarithmic expressions into simpler forms, especially when the calculator doesn’t support certain bases.
• It facilitates more straightforward numerical computation by using a base like 10, which is a standard on most calculators.

Understanding base conversion can make many logarithmic problems much more manageable and is a key skill in mathematical problem-solving.

Calculator Usage: Computing Logarithms

Using a calculator to compute logarithms, especially after using the change-of-base formula, is a valuable skill. Modern scientific calculators have built-in functions for base 10 (common logarithms) and base \( e \) (natural logarithms).
Here's how you can use a calculator effectively for logarithms:
1. **Locate the Logarithm Functions:** Most scientific calculators have \( \log \) and \( \ln \) buttons.
2. **Input Values for Base 10 Logarithm:** To calculate \( \log_{10} 175 \), enter 175, then press the \( \log \) button. Similarly, do this for \( 200 \).
3. **Compute the Division:** Once you have both values, divide them to find your result using the change-of-base formula:
\[ \frac{\log_{10} 175}{\log_{10} 200} \approx \frac{2.2430}{2.3010} \approx 0.974 \]This process allows you to handle logarithms beyond the standard bases directly, making complex calculations more accessible.
Pro tips for calculator use:

• Clear previous calculations to avoid errors in subsequent steps.
• Double-check entries for numerical accuracy.
• Familiarize yourself with your calculator’s specific functions to increase efficiency.

By mastering these steps, you can leverage your calculator to solve a wide array of logarithmic expressions efficiently.