Question

Use a calculator to evaluate each expression. Round your answer to three decimal places. $$ \frac{\ln \frac{2}{3}}{-0.1} $$

Short answer

4.055

Step-by-step solution

Identify the expression

The given expression to evaluate is \ \ \( \frac{\frac{\frac{2}{3}}}{-0.1} \).

Calculate the natural logarithm

Find the natural logarithm (ln) of \( \frac{2}{3} \) using a calculator. Ensure to enter \( \frac{2}{3} \) correctly. The calculation is \( \frac{2}{3} \approx 0.6667 \). Now, find \( \text{ln}(0.6667) \). This equals approximately \( -0.4055 \).

Divide by -0.1

Now divide the result from Step 2 by \( -0.1 \). Set up the division as follows: \( \frac{-0.4055}{-0.1} \). This equals \( 4.055 \).

Round the answer

Lastly, if needed, ensure the answer is rounded to three decimal places. The answer \( 4.055 \) is already in three decimal places.

Key concepts

natural logarithm

A natural logarithm (often abbreviated as 'ln') is a logarithm to the base of the mathematical constant \( e \), which is approximately equal to 2.71828. It helps us understand how many times we multiply \( e \) to get another number. For example, the natural logarithm of 1 (\( \ln(1) \)) is 0 because \( e^0 = 1 \). In contrast, the natural logarithm of \( e \) (\( \ln(e) \)) is 1 because \( e^1 = e \).

To solve for the natural logarithm of a fraction like \( \frac{2}{3} \):

• First, convert the fraction to its decimal form. Here, \( \frac{2}{3} \) is approximately 0.6667.
• Next, using a calculator, input \( \ln(0.6667) \). You’ll get about \( -0.4055 \).

It's crucial to use a calculator capable of natural logarithm calculations, especially when the numbers are not simple.

calculator usage

Calculators are essential for evaluating more complex expressions, such as natural logarithms. Most scientific calculators have a dedicated 'ln' button for finding natural logarithms. Here is a simple guide:

• Switch on your calculator and clear any previous data by pressing 'C' or 'AC'.
• Enter the number you want to find the natural logarithm for. For example, type in 0.6667.
• Press the 'ln' button. The display will show the natural logarithm of the entered number. In our case, it should be approximately -0.4055.

If your calculator doesn't have the 'ln' button, consider using an online calculator or a logarithm table.

With practice, calculator usage becomes a quick and effective way to handle complex calculations.

rounding numbers

Rounding numbers is a way to reduce the number of digits while keeping the number as close as possible to the original value. It is especially vital when working with long decimal places. For example, rounding to three decimal places means keeping only three digits after the decimal point.

• If the fourth digit after the decimal is 5 or greater, then increase the third digit by one.
• If the fourth digit after the decimal is less than 5, retain the third digit as it is.

In our exercise, the number \( 4.0555... \) rounds to 4.055 because the fourth decimal digit (in this case, 5) rounds up the third decimal.

It's always good practice to double-check your rounded results to ensure accuracy, especially in precise calculations.