Problem 7 - College Algebra

Question

Approximate each number using a calculator. Round your answer to three decimal places. \(e^{2.3}\)

Step-by-Step Solution

Answer

After performing these calculations, the result approximately equals \(9.974\).

Step 1: Calculate the exponential

Enter the expression \(e^{2.3}\) into a scientific calculator. 'e' is the base of the natural logarithm, and hence calculators have a specific button for it.

Step 2: Round the answer

Round the resulting number to three decimal places. In general, to do this, look at the digit in the fourth decimal place. If it is 5 or above, increase the digit in the third decimal place by 1. If it is less than 5, leave the digit in the third decimal place as it is.

Key Concepts

Understanding the Natural Logarithm

The natural logarithm is a mathematical function that relates to the concept of logarithms. Specifically, it uses the number "e" as its base. The number "e" is an irrational number approximately equal to 2.71828. The natural logarithm is often denoted as \(\ln(x)\), where "x" is the number you are taking the logarithm of. When using \(e\) in calculations, it is important to recognize it as the base of the natural logarithm, which is why many scientific calculators have a dedicated button for it.

• Exponential Function: When you see an expression like \(e^{x}\), it indicates an exponential function where "e" is raised to the power of "x".
• Natural Logarithm's Role: The natural logarithm is used to find the power to which "e" must be raised to obtain a particular number.

These concepts are widely used in calculus and mathematical models involving growth and decay processes, such as population growth and radioactive decay. Recognizing where and when to use the natural logarithm is key to solving exponential problems.

Using a Scientific Calculator

Scientific calculators are essential tools for solving mathematical problems involving advanced functions, such as exponentials and logarithms. Most scientific calculators come with a button labeled "\(e^{x}\)" or "exp" to help you calculate exponential functions. To compute \(e^{2.3}\), you would typically enter "2.3" and then press the "\(e^{x}\)" button.

Here are some tips for using a scientific calculator effectively:

• Getting Comfortable: Familiarize yourself with your calculator's functions and buttons. Read the manual if needed to understand the button layout and functions.
• Performing Calculations: Enter the number first, followed by pressing the "\(e^{x}\)" button to get the exponential result.
• Double-Checking: After entering your calculation, ensure the result matches your expectation, especially with powers and exponentials, since a small input error can lead to large output discrepancies.

Using your scientific calculator properly will aid in accuracy and efficiency, particularly with complex functions. Always remember to verify results when learning to ensure a solid understanding of the calculator's capabilities.

Rounding Numbers

Rounding numbers, especially in advanced mathematics, is vital for presenting data in a clear, concise manner. When working with scientific data or mathematical results, it is common to round numbers to a specified number of decimal places. In this context, rounding to three decimal places requires you to look at the fourth decimal digit after performing your calculations.

• Rounding Rule: If the digit in the fourth decimal place is 5 or greater, increase the third decimal place by 1. If it's less than 5, leave the third decimal place unchanged.
• Example: Suppose you get a result of 9.97456. To round this to three decimal places, you'd look at the fourth decimal digit (5) and round up, resulting in 9.975.

Rounding helps in maintaining consistency in the representation of data and simplifies the interpretation of results. It is a fundamental skill often required in mathematical problems, scientific studies, and everyday data handling.