Problem 24 - Intermediate Algebra

Question

For Problems \(21-30\), use your calculator to find each natural logarithm. Express answers to four decimal places. \(\ln 79.5\)

Step-by-Step Solution

Answer

\( \ln 79.5 \approx 4.3771 \)

Step 1: Understand the Problem

The problem is asking us to find the natural logarithm of the number 79.5 and express the result to four decimal places. The natural logarithm function is commonly represented as \( \ln \) and is understood to have a base of \( e \), where \( e \approx 2.71828 \).

Step 2: Use the Calculator

To find \( \ln 79.5 \), we need to use a scientific calculator that has a natural logarithm function. Enter the number 79.5 and press the \( \ln \) button to calculate its natural logarithm.

Step 3: Interpret the Result

After performing the calculation on the calculator, you will obtain a decimal number. In this case, the calculator provides the result for \( \ln 79.5 \) as 4.3771.

Step 4: Round to Four Decimal Places

The answer from the calculator is already expressed to four decimal places as 4.3771. Make sure that the fourth decimal digit is correct, and the number is appropriately rounded.

Key Concepts

Using a Scientific Calculator

A scientific calculator is an essential tool for calculating natural logarithms, as well as a wide range of mathematical functions beyond basic arithmetic. Unlike a standard calculator, a scientific calculator includes keys for operations such as square roots, exponents, and logarithms.

Understanding how to use a scientific calculator to find natural logarithms involves:

• Identifying the correct function key, usually labeled as "ln" for natural logarithms on most scientific calculators.
• Entering the value for which you need the logarithm, such as 79.5 in our original exercise.
• Pressing the "ln" button to execute the calculation and retrieve the result.

Mastering these steps ensures further calculations, like linear equations and exponential functions, can be effortlessly handled.

Understanding Base e

In mathematics, the base of the natural logarithm is the constant e, which is approximately 2.71828. The number e is a fundamental mathematical constant and serves as the natural base for logarithmic and exponential functions.

Here's what makes e special:

• It naturally occurs in various areas of mathematics, especially when dealing with growth or decay processes.
• The function \( e^x \) (the exponential function) is unique because its derivative is the same as the function itself, which is a key property used in calculus.
• Natural logarithms with base e are particularly useful in many scientific and engineering contexts due to these properties.

When working with logarithms, understanding e's role and its intrinsic connection to exponential growth patterns equips you to tackle complex mathematical problems.

Rounding Decimal Numbers

Rounding decimal numbers to a specified degree of accuracy is crucial in mathematics, especially when presenting solutions. Rounding regulates the number of digits and ensures clarity and precision. For instance, the result of \( \ln 79.5 \) is 4.3771, and it needs to be expressed to four decimal places.

How to round a decimal number:

• Identify the digit at the fourth decimal place. In 4.3771, this digit is 7.
• Look at the subsequent digit. If this digit is 5 or greater, round up the fourth digit by one. If it's less than 5, keep the fourth digit unchanged.
• With 4.3771, the rounding leaves the number unaffected as it is already at four decimal places.

Implementing these rounding techniques ensures the numerical results you present are correct and easy for others to understand. This practice is particularly important when reporting scientific results or engineering computations.