Problem 23 - Algebra 2

Question

Use a calculator to evaluate each expression to four decimal places. \(\ln 5.42\)

Step-by-Step Solution

Answer

The value of \(\ln 5.42\) is approximately 1.6896.

Step 1: Understand the Expression

The expression given is \(\ln 5.42\). This means we need to find the natural logarithm (base \(e\)) of the number 5.42. The natural logarithm is commonly represented by the function \(\ln\) and involves the constant \(e\) (approximately 2.71828).

Step 2: Use a Calculator

To solve \(\ln 5.42\) using a calculator, we locate the button or function for the natural logarithm, usually represented by \(\ln\). We then enter the number 5.42 into the calculator and press the \(\ln\) function key.

Step 3: Evaluate to Four Decimal Places

After entering the number and pressing the \(\ln\) key, the calculator will provide a decimal result. Make sure that this result is displayed to four decimal places. For example, if the calculator shows 1.68963159, round this number to 1.6896.

Key Concepts

Logarithmic Functions

Logarithmic functions help us in understanding growth processes and decay patterns. They transform multiplication into addition, making complex calculations easier. The natural logarithm, denoted by \( \ln \), is a type of logarithm found with the base \( e \). Here, the number \( e \) is a mathematical constant approximately equal to 2.71828. It's often used to model continuous growth, such as population or radioactive decay.A logarithm answers the question: To what power must we raise the base to obtain a certain number? For example, \( \ln 5.42 \) asks for the power to which \( e \) must be raised to get 5.42. This concept shows up in many fields, including mathematics, physics, and finance. By converting to the logarithmic form, various exponential functions can be simplified and analyzed.

Rounding Decimals

Rounding decimals is an essential skill for presenting data clearly. We reduce a number to fewer digits if all exact figures aren't needed. This process helps in simplifying results, especially when slight precision errors are acceptable.When rounding to four decimal places, we start from the fifth decimal and decide based on its value:

• If the fifth digit is 5 or higher, increase the fourth digit by 1.
• If it is less than 5, keep the fourth digit the same.

In our earlier example of calculating \( \ln 5.42 \), if the calculator shows 1.68963159, you round this to 1.6896. Rounding ensures that numbers stay precise enough for practical use while remaining easy to understand and work with.

Calculator Usage

Using a calculator effectively can significantly enhance your mathematical capabilities. It's handy for quickly solving expressions like \( \ln 5.42 \) without extensive manual calculation.To find the natural logarithm on a calculator:

• First, locate the \( \ln \) button. It's usually on scientific calculators.
• Next, input the number you wish to find the logarithm for. In our case, 5.42.
• Press the \( \ln \) button to get the result. Make sure the calculator displays enough decimals to allow accurate rounding.

Embrace familiarity with these functions to handle complicated expressions more efficiently and accurately solve problems. This practice improves both speed and confidence in tackling logarithmic calculations.