Problem 57 - Precalculus with Limits

Question

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. \(f(x)=\ln x \quad x=18.42\)

Step-by-Step Solution

Answer

After evaluating the function at the given value of \(x = 18.42\), the result is approximately 2.914.

Step 1: Understand the function

The given function is \(f(x)=\ln x\), which represents the natural logarithm of \(x\). We need to evaluate the function at the given value of \(x\) which is 18.42.

Step 2: Insert the value of \(x\) into the function

You will need to input the function into a calculator to evaluate \(\ln 18.42\).

Step 3: Calculate the function

Upon calculating the value of \(\ln 18.42\) using a scientific calculator, the value is approximately 2.914. Make sure to round this answer to 3 decimal places.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \(\ln x\), is a fundamental concept in mathematics, specifically in calculus and its applications. It refers to the logarithm base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. In simpler terms, if you have the equation \(\ln x = y\), it means that \(e^y = x\). This equation tells us how many times we must multiply \(e\) by itself to get the number \(x\).

Understanding natural logarithms is crucial because they describe growth processes, model decay in natural sciences, and are used in financial calculations like compound interest. In our exercise, we are interested in finding \(\ln 18.42\). This means we want to know the power to which \(e\) must be raised to yield 18.42.

Scientific Calculator Usage

Scientific calculators are invaluable tools for students and professionals in scientific, engineering, and mathematical fields. To evaluate natural logarithms using a scientific calculator, follow these steps:

• Turn on the calculator and check that it is in the correct mode for calculating logarithms.
• Locate the \(\ln\) button, which is the natural logarithm function key.
• Type in the value you want to find the natural logarithm of, in this case, 18.42.
• Press the \(\ln\) button after entering the number to display the result.
• If the calculator has a rounding function, you might need to round the answer manually according to the precision required by your exercise.

Always ensure your calculator's settings are correct for the mathematical operation you're performing to avoid errors.

Rounding Decimals

Rounding decimals is a mathematical process used to reduce the number of digits right of the decimal point. It simplifies numbers while maintaining a value that is very close to the original number. This is often necessary for real-world measurements where extreme precision is not required or when a simplified answer is easier to comprehend or work with.

To round to three decimal places, like in our exercise, follow these guidelines:

• Look at the fourth decimal place.
• If the fourth decimal place is 5 or more, increase the third decimal place by one.
• If the fourth decimal place is less than 5, leave the third decimal place as it is.

For example, if a number is 2.9136, rounded to three decimal places, it would be 2.914. It's also important to be aware of the context in which you're rounding because, in fields like banking or sciences, inaccurate rounding can lead to significant errors.