Problem 58 - Intermediate Algebra

Question

For Problems \(54-61\), perform the following calculations and express answers to the nearest hundredth. (These calculations are in preparation for our work in the next section.) $$ \frac{\ln 3}{0.04} $$

Step-by-Step Solution

Answer

27.47

Step 1: Understand the Expression

The problem requires calculating the expression \( \frac{\ln 3}{0.04} \), where \( \ln 3 \) denotes the natural logarithm of 3. We need to divide the natural logarithm of 3 by 0.04.

Step 2: Calculate \( \ln 3 \)

Find the natural logarithm of the number 3 using a calculator. \( \ln 3 \approx 1.0986 \).

Step 3: Perform the Division

Now divide the value of \( \ln 3 \) by 0.04. Compute \( \frac{1.0986}{0.04} = 27.465 \).

Step 4: Round to the Nearest Hundredth

Round the result from the division to the nearest hundredth. 27.465 rounded to the nearest hundredth is 27.47.

Key Concepts

Natural Logarithm

The natural logarithm, often represented as \( \ln \), is a special type of logarithm with the base \( e \), an irrational constant approximately equal to 2.71828. It's the power to which \( e \) must be raised to obtain a given number. The natural logarithm is particularly important in mathematics because it simplifies complex exponential expressions and frequently appears in calculus and other higher-level math operations.
If we take \( \ln 3 \) as an example, it means we're looking for the power to which \( e \) must be raised to get the number 3. Using a calculator, you can find that \( \ln 3 \approx 1.0986 \).
Natural logarithms are helpful in situations involving growth and decay, like financial calculations and scientific models. They also play a vital role in calculus, especially when dealing with integrals involving exponential functions.

Division of Real Numbers

Dividing real numbers follows the same basic principles as dividing any other type of number. Real numbers include all the numbers on the number line, such as whole numbers, fractions, and decimals. When you divide two real numbers, you're determining how many times the divisor can be subtracted from the dividend until nothing is left or the smallest remainder possible is achieved.
In the exercise \( \frac{\ln 3}{0.04} \), we take \( \ln 3 \), approximately 1.0986, and divide it by 0.04. Here's a step-by-step breakdown:

• Find \( \ln 3 \approx 1.0986 \)
• Perform the division: \( \frac{1.0986}{0.04} \) gets you 27.465

Division is crucial both in daily calculations and in understanding more complex mathematical concepts. It's essential to grasp this concept since it lays the foundation for more advanced operations.

Rounding Numbers

Rounding numbers is a common practice used to simplify figures to make them easier to work with while maintaining a degree of accuracy. It involves adjusting a number to reduce its number of decimal places based on a set rule.
When rounding to the nearest hundredth, you look at the third decimal place:

• If the digit in the third decimal place is 5 or higher, you round the number up.
• If it's less than 5, you round it down.

In the division of \( \frac{1.0986}{0.04} \), the result was 27.465. To round this to the nearest hundredth, observe the third decimal place, which is 5. Since 5 is equal to or more than 5, we round up, resulting in 27.47.
Rounding is important in statistics and when precision to many decimal places is unnecessary, helping to communicate results more clearly.