Problem 56 - College Algebra

Question

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. $$\ln \left(\frac{4}{5}\right)$$

Step-by-Step Solution

Answer

The rounded value of \(\ln\left(\frac{4}{5}\right)\) is approximately -0.223.

Step 1: Convert the Expression for Calculation

Start by writing the expression you want to evaluate: \( \ln \left( \frac{4}{5} \right). \) We'll input this into a calculator to find an approximate numerical value.

Step 2: Input the Expression into a Calculator

To calculate \(\ln \left(\frac{4}{5}\right)\), input \(\frac{4}{5}\) into the calculator first to get a decimal. That is, calculate \(4 \div 5 = 0.8\).

Step 3: Calculate the Natural Logarithm

Use the calculator to find \(\ln(0.8)\). This can be done by simply pressing the 'ln' button and typing in 0.8. The calculator will provide an answer to several decimal places.

Step 4: Round the Result

The result from the calculator for \(\ln(0.8)\) is approximately -0.223143551. Round this number to the nearest thousandth, which means keeping three decimal places: \(-0.223\).

Key Concepts

Rounding

Rounding is a numerical technique used to simplify figures by eliminating less significant digits. In mathematics, it allows you to represent numbers more simply without greatly affecting accuracy. Rounding to the nearest thousandth involves focusing on the digit in the thousandth place, which is three places to the right of the decimal point. After identifying the thousandth digit, you need to look at the digit immediately following it. If this digit is 5 or more, increase the thousandth digit by one. If it is less than 5, keep the thousandth digit unchanged. For instance:

• If you have -0.223143551, the thousandth digit is 3 and the next digit is 1, which is less than 5. Thus, you round it off to -0.223.
• For a number like 0.98762, the thousandth digit is 7, and since the next digit (6) is 5 or more, you round it up to 0.988.

Rounding helps in reporting numbers neatly, making them easier to comprehend and use in further calculations.

Calculator Use

A calculator is an essential tool for quick and accurate computation, especially when dealing with functions like the natural logarithm. It's crucial to understand the basic steps for correctly inputting and interpreting calculations. To calculate natural logarithms, most calculators have a specific 'ln' function, which directly computes this without requiring manual look-ups or conversions. Here's a simple guide to using a calculator for natural log calculations:

• First, ensure your calculator is in the correct mode, typically 'deg' or 'rad', depending on additional computations needed.
• Input the base value using either division or direct decimal input—like entering 0.8 when calculating \ln\(\frac{4}{5}\).
• Press the 'ln' button followed by the base value (e.g., 0.8) to get the logarithmic result.

These steps can help avoid errors and ensure you make the most of your calculator's capabilities. Understanding these functions not only simplifies homework tasks but is also useful in numerous real-world scenarios.

Expression Evaluation

Expression evaluation is the process of computing a numerical value from a mathematical expression. When expressions include natural logarithms, this involves several technical yet straightforward steps:To evaluate the expression \ln\(\frac{4}{5}\):

• First, you must simplify the fraction: \frac{4}{5} = 0.8\.
• Next, you input this decimal into the calculator's 'ln' function to compute the natural logarithm: \ln(0.8)\ is calculated as approximately -0.223143551.
• Finally, apply rounding rules to express it accurately to three decimal places: \approx -0.223\.

This structured methodology simplifies evaluation and ensures precision in results. Practice with such expressions can hone problem-solving skills, aiding in both academic and practical problem-solving.