Problem 55 - Algebra for College Students

Question

Perform the following calculations and express answers to the nearest hundredth. $$ \frac{\ln 2}{\ln 7} $$

Step-by-Step Solution

Answer

0.36

Step 1: Understand the Problem

The problem requires us to calculate \( \frac{\ln 2}{\ln 7} \) and round the result to the nearest hundredth. Here, \( \ln \) represents the natural logarithm, which is the logarithm to the base \( e \).

Step 2: Calculate \( \ln 2 \) and \( \ln 7 \)

We need the values of the natural logarithms of 2 and 7 to compute the quotient. Using a calculator, we find: \( \ln 2 \approx 0.6931 \) and \( \ln 7 \approx 1.9459 \).

Step 3: Compute the Ratio

Divide \( \ln 2 \) by \( \ln 7 \):\[ \frac{0.6931}{1.9459} \approx 0.3562 \].

Step 4: Round to the Nearest Hundredth

The result of the division is approximately 0.3562. Rounding this to the nearest hundredth gives us 0.36.

Key Concepts

Understanding Logarithmic Calculations

Logarithmic calculations involve understanding how to manipulate logarithmic expressions and using them in equations or calculations. The natural logarithm, denoted as \( \ln \), is particularly important because it uses the base \( e \), where \( e \approx 2.71828 \). In the exercise, when calculating \( \frac{\ln 2}{\ln 7} \), you're really working out the ratio of two logarithmic values. To do this, you must first know the logarithmic values of these numbers. In practice, this typically involves using a calculator since natural logarithms do not have simple fractional forms. Remember:

• Natural logarithms help in solving exponential decay and growth problems.
• They can convert multiplication into addition, simplifying complex calculations.
• Understanding logarithms is essential for higher-level math, engineering, and sciences.

The Importance of Rounding Numbers

Rounding numbers is a crucial skill, especially in mathematics. It is the process of adjusting a figure to make it simpler or to express it with less precision. This is often done to fit a specific requirement, such as expressing a result to the nearest tenth, hundredth, etc. In the exercise, after calculating \( \frac{\ln 2}{\ln 7} \approx 0.3562\), the result is rounded to the nearest hundredth to give \( 0.36 \). Here's why rounding is important:

• It increases clarity by reducing the chances of errors in calculation and communication.
• Ensures consistency, especially when figures are presented in reports or papers.
• Makes numbers easier to work with in mental math and approximations.

When rounding, if the digit after your rounding point is 5 or more, you round up. If it's less than 5, you round down. This rule helps maintain consistency.

Using Calculators in Algebra

Calculators are invaluable tools in algebra, assisting in performing complex calculations quickly and accurately. When you calculate natural logarithms, like \( \ln 2 \) and \( \ln 7 \), a scientific calculator or a calculator app is usually required, as these values aren't simple whole numbers. Calculators help by:

• Providing precise decimal values for logarithms and other functions.
• Handling complex equations effortlessly, reducing manual calculation errors.
• Offering additional functions like memory storage, which can be handy when dealing with longer calculations.

However, while calculators are useful, they should complement, not replace, an understanding of the underlying mathematical concepts. Understand what each button and function does, and practice solving problems without relying solely on calculators to strengthen your skills. This builds confidence and deepens comprehension in algebra and beyond.