Problem 27 - Algebra for College Students

Question

Use your calculator to find each natural logarithm. Express answers to four decimal places. \(\ln 0.46\)

Step-by-Step Solution

Answer

\( \ln 0.46 \) is approximately -0.7765.

Step 1: Understanding the Logarithm

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

Step 2: Calculating Using a Calculator

To find \( \ln 0.46 \), enter 0.46 into your scientific calculator and press the "ln" button. This will give you the natural logarithm of 0.46.

Step 3: Rounding the Result

The calculator provides the natural logarithm of 0.46 as approximately -0.7765288. Round this number to four decimal places to get -0.7765.

Key Concepts

Using Scientific Calculators

Using a scientific calculator can simplify the process of finding complex mathematical expressions, like the natural logarithm. These calculators have a specific button labeled "ln" for computing the natural logarithm, which is based on the mathematical constant \( e \). To calculate \( \ln 0.46 \) on your device, follow these simple steps:

• Power on your calculator, if necessary.
• Type "0.46" using the number keys.
• Locate and press the "ln" button.

Performing these actions will provide the natural logarithm of 0.46 immediately, removing the need for manual computation. This tool ensures accuracy, which is essential when dealing with precise values like those involved in logarithmic calculations.

Logarithm Rounding

Once you have calculated the natural logarithm using a scientific calculator, the next step is proper rounding. After finding \( \ln 0.46 \) and getting a result like -0.7765288, it’s important to round the number accurately to the desired decimal places.Usually, for homework and most math exercises, you'll be asked to round to a specific number of decimal places. Here we round to four decimal places. The correct process is:

• Look at the fifth number after the decimal point.
• If it's 5 or more, round the fourth decimal place up.
• If it's less than 5, keep the fourth decimal as is.

Following these rules, -0.7765288 rounds to -0.7765. Rounding correctly ensures precision, especially when the results are used in further calculations or real-world applications, where small discrepancies can lead to bigger issues.

Base e

The base \( e \) is a mathematical constant used primarily in logarithmic functions, like the natural logarithm. Known as Euler's number, \( e \) is approximately 2.71828 and arises naturally in various mathematical contexts, particularly those involving growth or decay, such as population models or financial calculations.When you encounter \( \ln \), it’s indicating the natural logarithm, which specifically uses \( e \) as its base. Here's why it's significant:

• It simplifies the differentiation and integration of exponential functions in calculus.
• Appears in compounding interest formulas, where continuous growth is modeled.
• Represents steady growth rates, making it invaluable in sciences and engineering.

Understanding \( e \) helps in grasping the natural logarithm’s role in mathematics, providing insight into why it functions as it does and how it is applied across different fields.