Problem 57 - Elementary and 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. $$\ln 1.72$$

Step-by-Step Solution

Answer

\( \ln 1.72 \approx 0.5416 \)

Step 1: Identify the Expression

The given expression is to find the natural logarithm of 1.72, which is represented as \( \ln 1.72 \). The natural logarithm (ln) refers to a logarithm with base \( e \), where \( e \approx 2.71828 \).

Step 2: Use a Calculator

To evaluate \( \ln 1.72 \) using a calculator, first turn on your calculator and locate the \( \ln \) function. This function is typically found among the keys or in a function menu. Enter 1.72 and press the \( \ln \) button to compute the value.

Step 3: Round the Result

After using the calculator, you will obtain a numerical result for \( \ln 1.72 \). The calculator displays this value with more than four decimal places. Round this result to four decimal places for the final answer.

Key Concepts

Understanding Base e

In mathematics, the concept of the natural logarithm revolves around a special number known as "base \( e \)." This number is approximately equal to 2.71828 and is unique because it is the foundation of natural exponential functions. Natural logarithms are denoted by \( \ln \) and differ from regular logarithms, which often have a base of 10. The use of \( e \) is significant in understanding exponential growth and decay. This includes processes in fields like biology, finance, and physics.
The number \( e \) emerges in many natural processes, which is why it plays a central role in the concept of natural logarithms. When you encounter \( \ln x \) in an expression, you're essentially determining the power to which you must raise \( e \) to obtain \( x \). For example, in the expression \( \ln 1.72 \), the calculator will find a value that, when used as an exponent on \( e \), results in 1.72.
To evaluate this without a calculator involves an understanding of exponential growth principles, but for most practical purposes, especially homework, a calculator does the trick!

The Art of Rounding Decimals

Rounding decimals is a fundamental skill in mathematics that ensures precision while presenting an answer in a manageable form. When dealing with expressions like \( \ln 1.72 \), calculators often provide results with many decimal places. Rounding is essential to give a cleaner, more practical number, suited for further calculations or recording in homework.
Here's how to round to four decimal places:

• Identify the fifth decimal digit; this helps decide whether to round the fourth decimal up or keep it the same.
• If the fifth decimal is 5 or more, increase the fourth decimal by one. If it's less than 5, the fourth decimal stays as is.

Thus, a number like 0.529376 rounded to four decimal places becomes 0.5294. Make sure to note the number in this rounded form, as this is the value required by most exercises like the one given.

Essential Calculator Usage

Using a calculator efficiently can seem daunting initially, but it's straightforward once you get the hang of it. Most scientific calculators have a dedicated \( \ln \) button because base \( e \) functions are so common in scientific calculations. Here are some steps you can follow to use your calculator for finding \( \ln \) values:

• Turn on the calculator and check for the \( \ln \) button. This is crucial for operations involving natural logarithms.
• Enter the number whose natural logarithm you wish to calculate—in our case, 1.72.
• Press the \( \ln \) button. The calculator will immediately compute the natural logarithm of your input.

Many calculators will present the answer with more digits than you need for your homework, which is why rounding is often necessary. Additionally, learning these steps will streamline your process and reduce errors, making it easier to handle more complex problems later on in your studies.