Problem 30 - Algebra for College Students

Question

Use your calculator to find each natural logarithm. Express answers to four decimal places. \(\ln 0.008142\)

Step-by-Step Solution

Answer

\( \ln(0.008142) \approx -4.8110 \).

Step 1: Identify the Function

The problem requires us to compute the natural logarithm of the number 0.008142. The natural logarithm function is denoted as \( \ln \) and has a base \( e \), where \( e \approx 2.71828 \).

Step 2: Input Value into Calculator

Enter the value 0.008142 into your calculator. Ensure your calculator is set to calculate natural logarithms (usually denoted with the "ln" button).

Step 3: Compute the Natural Logarithm

Press the "ln" button after entering the value to compute \( \ln(0.008142) \). The calculator will provide the result.

Step 4: Record the Result

The calculator should display a result approximately equal to \( -4.8110 \). This is the value of \( \ln(0.008142) \) rounded to four decimal places.

Key Concepts

Calculator Use in Mathematics

In mathematics, calculators can be incredibly useful tools for computing complex functions like the natural logarithm. They handle the computations quickly and accurately. To use a calculator for finding the natural logarithm, you should first ensure that your calculator is in the correct mode. Look for the button labeled "ln," which stands for the natural logarithm and is crucial when dealing with the base of Euler's number, \( e \). Calculators vary, so it's essential to become familiar with your specific device's functionality.

• Ensure it's set to compute natural logarithms by checking for the "ln" button.
• Enter the number precisely as it appears in your problem statement. In our case, it's 0.008142.
• Press the "ln" button after inputting the number to get the result.

This method efficiently determines values that would be cumbersome and time-consuming by hand. Using your calculator effectively in mathematics can save you a lot of effort and also ensure your results are accurate.

Rounding to Decimal Places

When working with calculations, rounding can clear up those long decimal numbers into more manageable figures. For this exercise, we round the calculated natural logarithm to four decimal places, ensuring consistent precision across all results. Rounding is straightforward:

• First, identify the digit at the specified decimal place. For four decimal places, look at the fifth decimal place to make decisions.
• If the digit is 5 or higher, increase the digit in the fourth decimal place by one. If it's less than 5, leave the fourth-place digit unchanged.
• So, if your calculator shows \( -4.810967 \), it’s rounded to \( -4.8110 \).

This type of rounding helps streamline your answers, making them easier to read and comparable with others' work to ensure consistency.

Exponential Function

The concept of an exponential function is integral to understanding natural logarithms. The base of the natural logarithm is the constant \( e \), approximately equal to 2.71828. This number is fundamental in the mathematical discipline, as it emerges in various growth-based processes.

• Natural logarithms answer the question: "What power must \( e \) be raised to, to produce this number?"
• In our exercise, finding \( \ln(0.008142) \) essentially asks what power of \( e \) gives us 0.008142.
• These calculations are vital in fields like compound interest, population dynamics, and even physics.

Understanding exponential functions provides a foundation for not only comprehending the logarithmic process but also for applications across scientific disciplines.