Problem 58 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. \(\ln (\sqrt{2})\)

Step-by-Step Solution

Answer

The value of \( \ln(\sqrt{2}) \) rounded to the nearest thousandth is 0.347.

Step 1: Understand the Problem

The exercise requires us to evaluate the expression \( \ln(\sqrt{2}) \) using a calculator and round the result to the nearest thousandth. The natural logarithm \( \ln(x) \) is the logarithm to the base \(e\), where \(e\) is approximately 2.71828. \( \sqrt{2} \) represents the square root of 2.

Step 2: Calculate the Square Root

First, calculate the square root of 2 using a calculator. \( \sqrt{2} \approx 1.4142136 \).

Step 3: Take the Natural Logarithm

Use a calculator to find the natural logarithm of the value obtained in Step 2. Calculate \( \ln(1.4142136) \). This yields approximately \( 0.34657359 \).

Step 4: Round the Result

The instruction is to round the result from Step 3 to the nearest thousandth. The number \( 0.34657359 \) rounded to the nearest thousandth is \( 0.347 \).

Key Concepts

Evaluating Expressions

When we talk about evaluating expressions, we mean finding what a mathematical expression represents as a number. In our exercise, the expression is \( \ln(\sqrt{2}) \). Let's break it down:- **\( \ln(x) \)**: The natural logarithm function, which tells us the power to which the base \( e \) (approximately 2.71828) must be raised to produce the number \( x \).- **\( \sqrt{2} \)**: The square root function, where we find a number that, when multiplied by itself, equals 2.The process of evaluating involves substitution and calculation:1. Compute the square root: Substitute \( \sqrt{2} \) with its decimal approximation, \( 1.4142136 \).2. Insert this approximation into the natural logarithm: \( \ln(1.4142136) \).By following these steps, you simplify the problem into manageable parts, making it easier to solve with a calculator.

Using a Calculator

Calculators are essential tools for evaluating complex expressions, especially those involving natural logarithms and square roots. Here's how you can use a calculator to evaluate \( \ln(\sqrt{2}) \):- **Find the Square Root**: Input 2 and use the square root function, often represented by a square root symbol (√) or a button reading 'sqrt' on your calculator. Record the result, which should be approximately \( 1.4142136 \).- **Evaluate the Natural Logarithm**: Input the calculated square root into the natural logarithm function, typically noted by 'ln' on the calculator. This will give you a value of around \( 0.34657359 \). Using a calculator minimizes errors and allows for quick computation, especially when dealing with transcendental numbers like \( e \). Just input the number step-by-step according to the calculator's operations, and check each step for accuracy.

Rounding Numbers

Rounding numbers is a fundamental skill for refining an answer to a specified degree of precision. In this case, we need to round to the nearest thousandth. Here's a simple guide:- **Identify the Thousandth Place**: In the decimal \( 0.34657359 \), the thousandth place is the third digit after the decimal, which is '6'.- **Look at the Next Digit**: The digit following '6', known as the rounding digit, is '5'.If the rounding digit is 5 or greater, you round up the thousandth place by one. Therefore:- Increase the 6 to a 7, resulting in \( 0.347 \).Rounding ensures numerical answers are not only precise but also easier to read and use in further calculations.