Problem 40 - Algebra and Trigonometry

Question

Use a calculator to evaluate the expression, correct to four decimal places. $$ \begin{array}{llll}{\text { (a) } \ln 27} & {\text { (b) } \ln 7.39} & {\text { (c) } \ln 54.6}\end{array} $$

Step-by-Step Solution

Answer

(a) 3.2958, (b) 1.9994, (c) 4.0009.

Step 1: Understanding Natural Logarithm

The expression \( \ln x \) is the natural logarithm of \( x \), which is the power to which the base \( e \) (approximately 2.71828) must be raised to produce \( x \). Our task is to calculate \( \ln 27 \), \( \ln 7.39 \), and \( \ln 54.6 \) using a calculator.

Step 2: Evaluate \( \ln 27 \)

Enter 27 into the calculator and then press the \( \ln \) button. You should get a value around 3.2958 when rounded to four decimal places.

Step 3: Evaluate \( \ln 7.39 \)

Enter 7.39 into the calculator and press the \( \ln \) button. The calculator should return approximately 1.9994 to four decimal places.

Step 4: Evaluate \( \ln 54.6 \)

Input 54.6 into the calculator and press the \( \ln \) button. The result should be approximately 4.0009 when rounded to four decimal places.

Key Concepts

Calculator Usage

Using a calculator can seem daunting at first, especially when dealing with functions like natural logarithms. However, with a little practice, it'll become second nature. Most scientific calculators have a specific button labeled "ln," which stands for natural logarithm. This function helps us find the logarithm of a number with base \( e \), approximately 2.71828. To use this function correctly, you need to:

• First, input the number for which you want to find the natural logarithm. For instance, if you're finding \( \ln 27 \), enter 27.
• Next, press the "ln" button. The calculator will instantly compute the natural logarithm of your number.

Remember, calculators are powerful tools that eliminate complex calculations, making learning much easier. As you work through problems, get comfortable using your calculator for quick and accurate results. It's also important to ensure your calculator is properly functioning and in the correct mode for the task at hand.

Rounding Decimals

Rounding decimals accurately is an essential skill when working with mathematical calculations, especially when using a calculator. Precise results sometimes must be rounded to a set number of decimal places to make the numbers easier to read and work with. When you need to round to four decimal places, it's crucial to:

• Identify the fourth digit after the decimal point.
• Check the digit immediately after this fourth digit. If it is 5 or more, increase the fourth digit by one, and if it is less than 5, leave the fourth digit unchanged.

For example, when rounding \( \ln 27 \) which gives you approximately 3.295836866, the fourth decimal place is 8, and the next digit is a 3, so it remains 3.2958. Being meticulous with rounding avoids small errors that could affect your final results.

Evaluating Expressions

Evaluating expressions with the help of a calculator involves a series of logical steps. When calculating natural logarithms like \( \ln 27 \), \( \ln 7.39 \), and \( \ln 54.6 \), break down the process into parts for clarity. First, understand what part of the expression you need to calculate, like \( \ln x \). Then:

• Use your calculator, starting with the necessary input and function, such as typing in 27 and pressing the "ln" button for \( \ln 27 \).
• Interpret the calculator's output and apply any required rounding.
• Write down your final answer to ensure you can move on confidently.

Remember, practice helps improve your speed and understanding of evaluating expressions. With repetition, you’ll become more efficient at breaking down and solving complex problems quickly and correctly.