Problem 58 - Intermediate Algebra

Question

Use a calculator to evaluate each expression, if possible. Express all answers to four decimal places. See Using Your Calculator: Evaluating Base-e (Natural) Logarithms. $$ \ln 2.7 $$

Step-by-Step Solution

Answer

\( \ln(2.7) \approx 0.9933 \).

Step 1: Understand the Expression

We want to evaluate the natural logarithm of the number 2.7. The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828.

Step 2: Set Up the Calculator

Make sure your calculator has a function to compute the natural logarithm. This is usually labeled as "ln". Ensure the calculator is in the correct mode for calculating logarithms if necessary.

Step 3: Input the Expression

Enter "\( \ln(2.7) \)" into your calculator. This can usually be done by pressing the "ln" button followed by typing "2.7".

Step 4: Evaluate on Calculator

Press the "Enter" or "=" button to compute the value. The calculator will return the result of \( \ln(2.7) \).

Step 5: Interpret the Result

The calculator gives a numeric value. Write down the result rounded to four decimal places. For \( \ln(2.7) \), the result should be approximately 0.9933.

Key Concepts

Understanding the ln Function

The "ln" function is a shorthand notation for the natural logarithm. It is a mathematical function that helps you find the power to which the base must be raised to produce a given number. The base used here is a special number called "e." The constant "e" is approximately equal to 2.71828. This unique base makes the natural logarithm extremely useful in various fields like calculus, physics, and finance. When you see "ln", think of it asking the question: "To what power do we need to raise the number e to get the number we have?" For example, when calculating \( \ln(2.7) \), we're determining the number that, when placed as the power of \( e \), equals 2.7. Remember, using the ln function is essential in solving problems involving exponential growth or decay.

Calculator Use for Natural Logarithms

Using a calculator to compute natural logarithms is quite simple, but your calculator must have the capability to perform this function. Most scientific and graphing calculators include an "ln" button. This button is specifically designed to handle natural logarithm calculations. Here’s a step-by-step guide on how to use the calculator:

• First, ensure that your calculator is turned on and ready for calculations.
• Check if you need to switch your calculator into the correct mode, often labeled as "scientific" or "logarithmic" mode.
• Press the "ln" button once you're ready to start the calculation.
• Type in the number you want to find the natural logarithm of. For this example, type "2.7".
• Press "Enter" or "=" to evaluate the expression.

These steps will give you the precise numerical value of the logarithm.

Base-e Logarithms Explained

Base-e logarithms, known as natural logarithms, utilize the irrational constant "e" as their base. This number "e" is approximately 2.71828 and is an important constant in mathematics, particularly in calculus. The beauty of using base-e in logarithms is due to the properties it provides when solving growth and decay problems. Natural processes, such as population growth and radioactive decay, are often modeled using the natural logarithm because their growth can be described as continuously compounded processes. Whenever you calculate a natural logarithm, remember you are working with one of the fundamental constants of mathematics. Its utility goes beyond complex functions; it is a natural fit for calculations involving continuous growth, making it invaluable in both theoretical and applied mathematics.

Rounding Decimals in Logarithmic Calculations

In mathematics, precise calculations are crucial, but they often involve long decimals that aren't practical for everyday use. When dealing with logarithms, especially in applications like natural logarithms, it's common to round the result to a fixed number of decimal places for convenience.For our task of evaluating \( \ln(2.7) \), the answer should be rounded to four decimal places, ensuring a balance between precision and practicality.Here are a few rounding tips:

• Look at the fifth decimal place to determine if you need to round up or down. If it is 5 or higher, increase the fourth decimal by one.
• Never simply cut off numbers without checking the fifth position. Rounding ensures the number remains as accurate as possible given the space restrictions.
• Write down the rounded answer clearly, especially in homework or reports, to convey exactly what the calculator displayed.

For \( \ln(2.7) \), the rounded result correctly expressed is approximately 0.9933.