Problem 22 - Intermediate Algebra

Question

For Problems \(21-30\), use your calculator to find each natural logarithm. Express answers to four decimal places. \(\ln 18\)

Step-by-Step Solution

Answer

\(\ln(18) \approx 2.8904\)

Step 1: Understand the Problem

The task is to find the natural logarithm of the number 18. The natural logarithm is denoted as \( \ln(x) \), where \( x \) is the number for which you need to find the logarithm. In this problem, \( x = 18 \).

Step 2: Use a Calculator

To calculate \( \ln(18) \) using a calculator, ensure your calculator is set to the correct mode for natural logarithms. Typically, there is a function or button labeled 'ln' specifically for this purpose. Enter 18 and press the 'ln' button.

Step 3: Record the Result

After calculating using the calculator, you obtain the result for \( \ln(18) \). Make sure to round this value to four decimal places as specified in the problem.

Key Concepts

Calculator Usage

Calculators are incredible tools for students, especially when learning about natural logarithms! Most scientific calculators have a specific button labeled 'ln' which is used for calculating natural logarithms. Here's a simple guide on using it:

• First, turn on the calculator and ensure it is in the correct mode for decimal calculations.
• Next, find the 'ln' button on your calculator. It's usually on the left-hand side, either directly accessible or as a secondary function.
• Then, type in the number whose natural logarithm you want to find, in this case, 18.
• After entering the number, press the 'ln' button.
• Finally, read the result displayed on the screen. Don't worry if it’s a long decimal; you’ll round it in the next step!
  

It's easy and fast with a calculator, saving time from manual calculations, and allowing for accurate results.

Rounding Decimal Places

Rounding your final answer to four decimal places makes it precise enough for most academic settings without being overly exact.
To round correctly, follow these simple steps:

• Start with the number you've calculated, for example, 2.8903718 for \({ln(18)}\).
• Identify the digit at the fourth decimal place. In this case, it is '3'.
• Look at the next number (the fifth decimal place), which is '7'.
• Since '7' is greater than or equal to 5, you round '3' up to '4'.
• Thus, your answer becomes 2.8904.

Rounding helps in maintaining uniformity in reporting results and avoiding unnecessary detail in written work.
It's important to remember this rule: If the digit immediately after the desired precision is five or more, round up!

Finding Logarithms

Natural logarithms are based on the constant \( e \), which is approximately equal to 2.71828.
They are used commonly in sciences and mathematics to simplify complex calculations. When you see \( {ln(x)}\), it's asking you to find the power you need to raise \( e \) to get \( x \).

Here's a simple way to understand it:

• The natural logarithm \( {ln(18)} \) is asking: 'What power must \( e \) be raised to in order to equal 18?'
• Using a calculator, the answer approximately turns out to be 2.8904.

This means \( e^{2.8904} \approx 18 \).
Logarithms transform multiplicative relationships into additive ones, making calculations simpler, especially with exponential growth processes like interest, population growth, and decay.
Given their widespread application, mastering logarithms is essential for students!