Problem 54 - 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. $$\ln 0.675$$

Step-by-Step Solution

Answer

The value of \( \ln 0.675 \) is approximately -0.3920.

Step 1: Identify Expression

The task is to evaluate the natural logarithm of 0.675 using a calculator. The expression given is \( \ln 0.675 \).

Step 2: Open Calculator

Access a calculator that is capable of evaluating natural logarithms. Most scientific calculators have this function as a button marked 'ln'.

Step 3: Enter the Expression

On the calculator, input '0.675' and then press the 'ln' button to compute the natural logarithm of the number.

Step 4: Obtain the Result

After calculating \( \ln 0.675 \), the calculator displays the result. Make sure the result is displayed to four decimal places.

Step 5: Record the Final Answer

Note down the result. For \( \ln 0.675 \), the value to four decimal places is approximately -0.3920.

Key Concepts

Scientific Calculator

A scientific calculator is an essential tool for solving mathematical problems that involve complex computations, like evaluating natural logarithms. These calculators are equipped with advanced functions beyond the basic arithmetic calculations, specifically designed to handle scientific notation and logarithmic functions.

Most scientific calculators feature a dedicated button labeled 'ln' that specifically calculates the natural logarithm, which is the logarithm to the base 'e'. This button simplifies the process by allowing users to directly input decimal numbers and obtain the natural logarithm without manual calculations.

• Ensure your calculator is in the correct mode (usually the standard calculation mode is suitable).
• Locate the 'ln' button, which stands for logarithm base 'e'.
• These calculators often also have other logarithmic functions, like log (base 10), which might be marked differently.

Evaluate Logarithms

Evaluating logarithms using a calculator involves a few easy steps that enable you to find the logarithm of any number, such as the natural logarithm (ln).

When you are given a task like evaluating \( \ln 0.675 \), you start by inputting the number into the calculator.

• Enter the number (0.675, in this case) into the calculator.
• Press the 'ln' button, allowing the device to perform the calculation to find the natural logarithm of that number.
• The calculator then processes and displays the result without you having to do any manual computations.

Remember that the 'ln' button specifically helps you find the logarithm with base 'e', a mathematical constant approximately equal to 2.71828. This method is straightforward and helps you avoid manual errors, especially when handling more complex numbers.

Decimal Places

When expressing results in mathematics, precision is often crucial. This is where decimal places come into the picture, ensuring clarity and accuracy in your answers.

After you evaluate a natural logarithm, for example, \( \ln 0.675 \), you'll typically want to round the result to a certain number of decimal places. Depending on the instructions or requirements you're following, you might need to round the answer to four decimal places, as shown in this exercise.

• First, understand what rounding to four decimal places means: keeping four digits following the decimal point.
• Identify which digit would be the fourth position and check the next digit to see if you round up or not.
• For \( \ln 0.675 \), the answer of -0.3920 is already rounded to four decimal places, which is essential for consistent and precise results.

Rounding correctly is vital in fields requiring high precision, such as engineering and science, ensuring your calculations are both consistent and reliable.