Problem 51 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to evaluate the logarithm. Round your result to three decimal places.\(\ln 18.42\)

Step-by-Step Solution

Answer

The natural logarithm of \(18.42\), rounded to three decimal places, varies slightly depending on the calculator used, typically falling approximately around \(2.915\). It's recommended to check with a calculator for the exact value.

Step 1: Identify the log base

Here, the base is identified to be \(e\), which means we are dealing with natural logarithms. The symbol \(\ln\) is usually used for natural logarithms.

Step 2: Use a calculator

Use a scientific calculator. Input the number \(18.42\) into the \(\ln\) function of the calculator.

Step 3: Round the result

After you get a result from the calculator, round off the value to the nearest thousandth place, which is three decimal places.

Key Concepts

Logarithm Rounding

Rounding a logarithmic result is a common task in math exercises, especially when dealing with decimals. Here's a simple way to make it easy:
Start by identifying that you're working with a decimal number. In this exercise, after calculating the natural logarithm of 18.42, you'll have a decimal result.
Rounding to three decimal places means you keep three digits to the right of the decimal point.

• Look at the fourth digit (right after your third decimal place).
• If it's 5 or more, add 1 to the third decimal place.
• If it's less than 5, leave the third decimal as it is.

So, when rounding, precision is key, and following these steps will help ensure accurate results.

Scientific Calculator Use

Using a scientific calculator to find a natural logarithm is straightforward. Here's how you can do it:
First, make sure you're familiar with your calculator's functions. Look for the button labeled "ln", which stands for natural logarithm.

• Turn on your calculator.
• Enter the number, in this case, 18.42.
• Press the "ln" button to perform the operation.

You'll get a numerical result on your display. Using a scientific calculator ensures that complex calculations are done quickly, giving you more time to understand the concepts. Don't forget to practice using your calculator to become more comfortable with its functions.

Base e Identification

In natural logarithms, the base is the mathematical constant e, approximately equal to 2.718. Recognizing when to use base e can simplify your calculations.
The symbol "ln" is commonly used to indicate a natural logarithm, which implicitly uses base e.
Understanding this distinction is important because it differentiates natural logs from common logs, which use base 10.

• Natural logs are used extensively in fields like calculus and complex number theory.
• Knowing when to use "ln" helps in solving problems related to growth and decay processes, such as population growth or radioactive decay.

Identifying base e correctly ensures accurate logarithmic calculations, crucial for various mathematical applications.