Problem 59 - Algebra for College Students

Question

Perform the following calculations and express answers to the nearest hundredth. $$ \frac{\ln 2}{0.03} $$

Step-by-Step Solution

Answer

The answer is 23.10.

Step 1: Identify the expression

We have the mathematical expression \( \frac{\ln 2}{0.03} \). This means we need to calculate the natural logarithm of 2 and then divide the result by 0.03.

Step 2: Calculate \( \ln 2 \)

The natural logarithm of 2 is calculated using a calculator or logarithm tables. It is approximately \( \ln 2 \approx 0.693147 \).

Step 3: Perform the division

Divide the result of \( \ln 2\) by 0.03: \[ \frac{0.693147}{0.03} \] Use a calculator to perform the division.

Step 4: Round the result

The division \( \frac{0.693147}{0.03} \approx 23.1049 \). Round this result to the nearest hundredth, which gives \( 23.10 \).

Key Concepts

Logarithm Tables

Logarithm tables are a fascinating tool from the past, often used before calculators became widespread. They allowed people to easily find logarithmic values, which are essential in various mathematical calculations, especially those involving exponential growth or decay. In these tables, the logarithm of a number is usually provided with a specific base, such as 10 or Euler's number \(e\), used in natural logarithms.

For our calculation involving \( \ln 2 \), a logarithm table could show the value directly or help you locate a close approximation by looking up similar entries. However, with the advent of digital technology, calculators and computers are now primarily used due to their speed and accuracy. Yet, understanding how to read and use these tables fortifies your grasp of logarithmic concepts and their history in mathematics.

Division

Division is one of the four basic arithmetic operations and involves splitting a number into equal parts. In mathematical terms, it finds how many times one number, called the divisor, can fit into another, the dividend. In our problem, after calculating the natural logarithm of 2, which is approximately 0.693147, division helps us distribute this value evenly by 0.03.

When we divide \( \ln 2 \) by 0.03 using a calculator, the operation looks like \( \frac{0.693147}{0.03} \). Here, 0.03 is the divisor, representing the size of each "part." The process of division requires precision, especially when dealing with non-integer numbers, as rounding errors can emerge. Calculators are an invaluable tool here, providing a means to obtain precise results quickly, thus minimizing rounding issues during such calculations.

Calculator Use

Calculators are powerful tools that help solve complex mathematical problems with ease and speed. Whether you are finding logarithms or performing divisions, calculators are indispensable. Particularly when working with natural logarithms and small decimal divisors, their precision is unmatched.

To use a calculator for finding \( \ln 2 \), you key in 2 and then use the logarithm function, usually labeled "ln" on scientific calculators. This swiftly gives you approximately 0.693147. Following this, to divide by 0.03, you simply enter the division operation followed by 0.03. Calculators do the heavy lifting, presenting you with an exact decimal result, which in this instance is approximately 23.1049.

• Ensure your calculator is correctly set up for natural logarithms.
• Remember to round the final decimal answer as necessary, ensuring accuracy in your results.
• Recheck calculations to avoid input errors, particularly with small decimal values.

By efficiently using a calculator, you significantly reduce the risk of mistakes and can focus more on understanding concepts rather than getting caught up in intricate calculations.