Problem 53 - Elementary and Intermediate Algebra

Question

Use a calculator to evaluate each expression, if possible. Express all answers to four decimal places. See Using Your Calculator: Evaluating Base-e (Natural) Logarithms. $$\ln 35.15$$

Step-by-Step Solution

Answer

The value of \( \ln 35.15 \) is approximately 3.5581, rounded to four decimal places.

Step 1: Understand the Function

The given expression is \( \ln 35.15 \), which involves the natural logarithm function. The natural logarithm means we are looking for the power to which the base \( e \approx 2.718 \) must be raised to yield 35.15.

Step 2: Use the Calculator

On your calculator, find the button labeled 'ln' or 'natural log.' Enter 35.15 into your calculator and press the 'ln' button to compute the natural logarithm of 35.15.

Step 3: Interpret the Result

After pressing the 'ln' function, your calculator should display a numerical value that represents \( \ln 35.15 \). Make sure your calculator is set to display at least four decimal places so that you get an accurate result as requested.

Step 4: Round the Value

Round the result from the calculator to four decimal places to express the logarithm as requested. This will ensure that your answer is precise and aligns with the problem's instructions.

Key Concepts

Understanding the Base-e Logarithm

The natural logarithm, denoted as \( \ln \), is a special kind of logarithm that uses the number \( e \) as its base. The number \( e \), approximately equal to 2.718, is one of the most important constants in mathematics, similar to \( \pi \). When you see \( \ln 35.15 \), it means you are determining the exponent such that \( e \) raised to that power equals 35.15. To put it simply, if \( e^x = 35.15 \), then \( x = \ln 35.15 \). This concept is widely used in fields like calculus and financial calculations because of its close relationship with growth processes and continuous compounding. Understanding the base-e logarithm is critical because it forms the foundation for exponential growth equations and solving problems involving continuous change.

Calculator Use in Mathematics

Calculators are essential tools in mathematics, providing support for complex calculations that would otherwise take a significant amount of time. When dealing with natural logarithms, the 'ln' button on your calculator is crucial. Here's how to use it for evaluating \( \ln 35.15 \):

• First, turn on your calculator and find the 'ln' button, usually located alongside other logarithmic and mathematical functions.
• Enter the number 35.15 first. It’s important to ensure preciseness by typing the number accurately.
• After entering the number, press the 'ln' button. This function will compute the natural logarithm of the number you entered.

You will get an answer displayed on the screen, which represents the natural logarithm. Make sure to familiarize yourself with the specific functions of your calculator, as they might differ slightly based on the model. This familiarity will make your mathematical tasks quicker and more efficient.

Rounding Decimal Places

Rounding to decimal places is a key step in ensuring the accuracy and consistency of your mathematical solutions. When the instruction asks you to round to four decimal places, you must:

• Look at the fifth decimal place to decide whether to round up or stay the same. If the fifth digit is 5 or greater, increase your fourth decimal by one; otherwise, it remains unchanged.
• A number such as 3.857942 would round to 3.8579 when rounded to four decimal places.
• Always check your calculator's settings to make sure it shows enough decimal places so you can apply the proper rounding techniques.

Rounding is not just about cutting off numbers; it's about maintaining the integrity of your operations and providing results that are both precise and easy to work with. This practice is crucial, particularly in scientific and financial calculations, where precision significantly impacts outcomes.