Problem 58 - Algebra for College Students

Question

Perform the following calculations and express answers to the nearest hundredth. $$ \frac{\ln 3}{0.04} $$

Step-by-Step Solution

Answer

27.47

Step 1: Understand the Task

The task is to compute \( \frac{\ln 3}{0.04} \) and round the result to the nearest hundredth. "ln" denotes the natural logarithm, which is the logarithm to the base \( e \).

Step 2: Calculate \( \ln 3 \)

Use a calculator to find \( \ln 3 \). It is approximately 1.0986. This value will be used in the next step of calculation.

Step 3: Divide \( \ln 3 \) by 0.04

Now, divide the calculated \( \ln 3 \) which is approximately 1.0986 by 0.04: \[ \frac{1.0986}{0.04} = 27.465 \]

Step 4: Round to the Nearest Hundredth

The result from Step 3 is 27.465. Now, round this value to the nearest hundredth. Since the third digit after the decimal is 5, round up to 27.47.

Key Concepts

Rounding to Nearest Hundredth

Rounding numbers helps simplify them, making them easier to understand. When you round a number to the nearest hundredth, you're deciding which value is closest to the actual number, with two decimal places. To do this, you look at the third decimal place.

• If the third decimal is 5 or more, you round up.
• If it's less than 5, you round down.

For example, consider the number 27.465. Since the third digit (5) is 5 or more, you increase the second digit (6) by one. So, 27.465 rounded to the nearest hundredth becomes 27.47.
Rounding is helpful in making numbers more manageable and is often used when you need a quick estimate or when dealing with money.

Division in Algebra

In algebra, division is about splitting numbers into equal parts or groups. When you divide one number by another, you're finding how many times the divisor fits into the dividend.
Consider the expression \( \frac{a}{b} \), where \( a \) is the dividend, and \( b \) is the divisor. In our example, \( a \) is \( \ln 3 \) and \( b \) is 0.04. The goal is to determine how many times 0.04 fits into \( \ln 3 \).
In algebraic division, you can also encounter variables and constants, but for our example, using a calculator simplifies the calculation:

• First, compute or find the value of \( \ln 3 \).
• Then, divide that value by 0.04.

The solution shows that \( \frac{1.0986}{0.04} = 27.465 \). Breaking problems into steps allows for easy calculations and understanding.

Using a Calculator for Logarithm

Calculators are powerful tools for solving logarithms quickly. The natural logarithm \( \ln \), found on most scientific calculators, is the logarithm to the base \( e \). This is useful because calculating logarithms by hand is complex.
To find \( \ln 3 \) using a calculator:

• Turn on your scientific calculator.
• Find the "ln" button, which may vary in placement based on the model.
• Press the "ln" button and then enter the number 3.
• Hit the "equals" key or press "enter."
• The calculator should display approximately 1.0986.

Using calculators makes calculating natural logarithms fast and efficient, freeing you to focus on analyzing results rather than complex calculations. Always ensure your calculator is set to the correct mode if you're combining operations like division.