Problem 57 - 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

\( \ln 1.72 \approx 0.5414 \).

Step 1: Identify the Expression to Evaluate

The expression given is \( \ln 1.72 \). This represents the natural logarithm of 1.72.

Step 2: Understand Natural Logarithms

The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \). The constant \( e \) is approximately 2.71828.

Step 3: Use a Calculator

Using a scientific calculator, input the expression \( \ln 1.72 \). Ensure your calculator is set to the correct mode to handle natural logarithms.

Step 4: Evaluate and Round the Result

After calculating \( \ln 1.72 \), the result is approximately 0.5414. Round your answer to four decimal places, if necessary.

Key Concepts

Understanding Base-e Logarithms

To understand base-e logarithms, one must first be introduced to the concept of logarithms themselves. A logarithm indicates the power to which a base number must be raised to obtain a particular value. In the case of base-e logarithms, often known as natural logarithms, the base is the mathematical constant \( e \), which is roughly equal to 2.71828. So, when you see \( \ln 1.72 \), it is asking you to find the power that \( e \) must be raised to in order to get 1.72. This specific type of logarithm is common in mathematics and science because \( e \) naturally appears in many growth processes and exponential functions.
Natural logarithms are denoted by \( \ln \), as opposed to \( \log \), which is often used for base-10 logarithms. Recognizing the base is crucial when solving logarithmic expressions to ensure proper calculation and understanding.

Using a Scientific Calculator for Base-e Logarithms

Calculating a natural logarithm like \( \ln 1.72 \) is straightforward with a scientific calculator. Here are the steps to evaluate natural logarithms with precision:

• First, ensure that your calculator is in the correct mode. Not all calculators default to base-e logarithms, so locate the \( \ln \) button, which is specifically for natural logarithms.
• Enter the number whose natural logarithm you want to find. For example, type "1.72" into your calculator.
• Press the \( \ln \) button to perform the calculation. The calculator may require different order entries, so familiarize yourself with the function, possibly by consulting the calculator's manual.

This process will compute \( \ln 1.72 \), which allows you to compare its magnitude and understand relationships between exponential growth rates and other phenomena naturally described by \( e \).

Importance of Rounding Decimal Places in Calculations

Rounding is a critical step in reporting numerical results accurately yet succinctly. After calculating \( \ln 1.72 \) on your scientific calculator, the unrounded result may appear as a long decimal. For most academic and practical applications, it is standard to round this result to four decimal places. Here’s why:

• Clarity and Precision: Reporting results to four decimal places ensures that the number is precise enough for high accuracy while remaining clear and readable.
• Consistency: Using a consistent number of decimal places, such as four, provides uniformity across calculations, making data easier to compare and interpret.
• Practicality: In many scientific fields, extreme precision is not necessary, and overly extended decimal places may not add value to the result interpretation.

Therefore, the rounded result of \( \ln 1.72 \), approximately 0.5414, satisfies both clarity and accuracy, making it suitable for use in further computations or analytical processes.