Problem 52 - A Graphical Approach to College Algebra

Question

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\ln 1247$$

Step-by-Step Solution

Answer

\( \ln 1247 \approx 7.129 \)

Step 1: Understanding the Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a mathematical function that represents the logarithm to the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. We want to compute \( \ln 1247 \), which is asking what power we need to raise \( e \) to in order to get 1247.

Step 2: Use a Calculator to Compute \( \ln 1247 \)

To find \( \ln 1247 \) using a calculator, you will typically press the "ln" button followed by typing 1247 and then pressing the equals button (if needed). Make sure your calculator is set to the correct mode for natural logarithm calculations.

Step 3: Interpret the Result

Once you enter \( 1247 \) into the calculator using the natural logarithm function, you will receive a decimal approximation. The result should be approximately 7.129.

Key Concepts

Decimal Approximation

Decimal approximation is the process of representing a number in a simpler form, usually by rounding it to a manageable number of decimal places. This is especially useful when dealing with irrational numbers, such as logarithms, which often result in non-terminating decimals that can be cumbersome to work with. When you compute a logarithm like \( \ln 1247 \), the result is often a long decimal. In practice, we usually round this number to a set number of decimal places for convenience. For instance, \( \ln 1247 \) is approximately 7.129, when rounded to three decimal places. This means that the value has been simplified to reflect three numbers after the decimal point. Using decimal approximations helps in:

• Making complex calculations more manageable.
• Ensuring that the numbers are easier to read and interpret.
• Saving space and time in writing out long numbers.

Remember, while decimal approximations make our work easier, they do involve a small loss of precision.

Calculator Usage

Using a calculator simplifies finding the natural logarithm of a number, especially one as large as 1247. Calculators are designed to handle complex operations like logarithms with ease, ensuring accuracy in the process.To find \( \ln 1247 \) using a calculator, follow these general steps:

• Locate the "ln" function on your calculator. This might be a stand-alone button or part of a menu, depending on the model you are using.
• Enter the number, in this case, 1247, after selecting the ln function.
• Press "equals" ("=") or "enter" to get the result.

Make sure your calculator is set to the appropriate mode, which is usually standard for logarithms. Using the calculator for logarithms is not only fast but also improves:

• Accuracy, reducing the chance for human error in manual computation.
• Speed, allowing you to quickly solve complex mathematical problems.
• Efficiency, enabling you to focus on interpreting results rather than computing them manually.

If you ever get a result that doesn't seem right, double-check your input, and make sure the calculator is functioning correctly.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as fractions or ratios of integers. They have non-terminating, non-repeating decimal expansions. Examples include \( \pi \), \( \sqrt{2} \), and the mathematical constant \( e \), which is approximately equal to 2.71828.The natural logarithm involves \( e \), which means that results like \( \ln 1247 \) are often irrational. Because \( e \) is irrational, when you ask what power you need to raise \( e \) to get to 1247 (in potential terms of \( \ln 1247 \)), the answer is also an irrational number. This is why you end up with a decimal approximation.Understanding irrational numbers is key because:

• They help us explore and understand continuous growth or decay in real-world phenomena.
• They serve as the basis for more complex mathematical constructs like calculus and mathematical modeling.
• They remind us that not all numbers fit neatly into simple ratios or fractions, expanding our understanding of mathematics.

The beauty of irrational numbers is in their infinite complexity, showcasing the richness of mathematics beyond basic arithmetic.