Question

Use a calculator to evaluate each expression. Round your answer to three decimal places. $$ \frac{\ln \frac{10}{3}}{0.04} $$

Short answer

30.1

Step-by-step solution

Evaluate the natural logarithm

First, calculate the natural logarithm of the fraction \(\frac{10}{3}\). Use a calculator to find \(\frac{10}{3} \approx 3.333\). Then compute \(\frac{\text{ln}(3.333)}\) to get \(\text{ln}(3.333) \approx 1.204\).

Divide by the given number

Next, divide the result from Step 1 by 0.04. That is, compute \(\frac{1.204}{0.04} \approx 30.1\).

Round to three decimal places

Finally, make sure the result is rounded to three decimal places. Since 30.1 is already to one decimal place, it remains the same after rounding to three decimal places.

Key concepts

Natural Logarithm

The natural logarithm, denoted as \(\text{ln}\), is a logarithm with the base 'e'. The number 'e' is an irrational constant approximately equal to 2.71828. In simpler terms, the natural logarithm of a number is the power to which 'e' must be raised to produce that number. For instance, if you want to find the natural logarithm of \(\frac{10}{3}\), it's like asking, 'To what power must 'e' be raised to get 3.333?' \(\text{ln}(3.333) \) gives this value.
To compute the natural logarithm of a number such as \(\frac{10}{3}\), you first divide 10 by 3 to get approximately 3.333. Then, apply the natural logarithm function to this result: \(\text{ln}(3.333) \). This step is essential in many mathematical and practical problems.
In our exercise, \(\text{ln}(3.333) \) approximately equals 1.204.

Calculator Usage

When calculating natural logarithms and other advanced functions, a scientific or graphing calculator significantly simplifies the process.
Here's how to use a calculator for finding the natural logarithm and performing the division:

• First, enter the number you want to find the natural logarithm of, in our case, 3.333.
• Next, press the 'ln' button to obtain the natural logarithm. The calculator should display approximately 1.204.
• Then, enter this result and divide it by the desired number, which in our problem is 0.04.
• Press the '=' button to get the final result, approximately 30.1.

Using a calculator ensures accuracy and saves time. Make sure your calculator is in the correct mode, if applicable, to prevent any errors.

Rounding Decimals

Rounding decimals is an essential skill in mathematics, especially when dealing with results from complex calculations. Rounding helps to make numbers simpler and easier to work with, without needing extreme precision.
Here’s a quick guide on rounding to three decimal places:

• Identify the third decimal place. In the number 30.1, since there are no digits after the first decimal place ('1'), it remains as-is when rounded to three decimal places (30.100).
• If the digit right after the third decimal place (the fourth digit) is 5 or greater, add 1 to the third decimal place.
• If it’s less than 5, simply leave the third decimal place as-is.

For example, let’s say you have the number 1.6789:

• The digit in the third decimal place is '8'.
• Look at the fourth digit, which is '9'.
• Since '9' is greater than '5', round up the third decimal place from '8' to '9'.
• Your rounded number will be 1.679.

In our given problem, the number 30.1 remains the same after rounding to three decimal places because there are no additional decimal places to consider beyond the first.