Problem 42 - Precalculus : Building Concepts and Connections

Question

Evaluate the expression to four decimal places using a calculator. $$-\ln \frac{2}{3}$$

Step-by-Step Solution

Answer

To answer this question, you'll need to rely on your calculator. By entering -ln(2/3) and rounding to four decimal places, you'll find the correct solution to the exercise.

Step 1: Identify the Function

Recognize that the \(-\ln \) symbolizes the negative natural logarithm. This function is found on most scientific calculators.

Step 2: Determine the Value

Enter \( \frac{2}{3} \) into your calculator. Then, find the \(\ln \) or \(\log_e\) button on your calculator, which stands for the natural logarithm, and apply it to the result.

Step 3: Apply the Negation

Take the negative of the result because there is a minus sign before the \(\ln \) in the original expression.

Step 4: Round the Result

Depending on your calculator, the result can have many trailing digits. Only the first four decimal places are needed according to the exercise. Use standard rounding rules.

Key Concepts

Scientific Calculator Usage

When evaluating expressions that involve natural logarithms, a scientific calculator is an essential tool. These calculators have specific buttons or functions that allow you to calculate the natural logarithm, commonly denoted as \(\) or \(\log_e\). First, you'll need to power on your calculator. Then, input the fraction or decimal value you're working with using the appropriate numeric keys. In this exercise, we input \(\frac{2}{3}\). Afterward, locate the \(\ln\) button. This button might be under multiple layers of functions on more advanced calculators, sometimes accessed through a secondary key like 'shift' or '2nd'. Simply press this button after entering your number to get the logarithmic result. Understanding this function is key in evaluating expressions accurately.

Function Evaluation

Function evaluation refers to the process of finding the value of a function for a particular input. In this context, we are evaluating the natural logarithm function, \(\ln(x)\). This function determines the power to which the base number \(e\) (approximately 2.718) must be raised to yield \(x\). So, for \(\frac{2}{3}\), you will input it in your calculator, press the \(\ln\) button, and the calculator will compute the logarithm for you. This result represents the logarithmic scale of the original fraction with respect to the base \(e\). It's essential to correctly interpret the calculator's output and ensure you're evaluating the desired function as per your expression.

Rounding Rules

When working with calculators, results often exceed the precision required for your exercise. Here lies the importance of rounding rules. In this example task, the final result should be rounded to four decimal places. Rounding means adjusting the digits of a numerical value to make it simpler while retaining its value close to the original.
Here's how you round to four decimal places:

• Identify the fourth decimal place.
• Look at the fifth decimal place, which will determine if you round up or stay the same.
• If the fifth digit is 5 or higher, increase the fourth digit by one.
• If the fifth digit is below 5, leave the fourth digit unchanged.

Following these steps ensures the result remains accurate and concise for your needs.

Mathematical Negation

Mathematical negation involves modifying the sign of a number. In the context of this exercise, you're required to negate the natural logarithmic result. This is represented by the minus sign before \(\ln\), which indicates you should take the opposite sign of the ln value calculated. Simply, once you've found the \(\ln\) value using your calculator, switch its sign by applying a negative sign. Most scientific calculators have a \(-\) button or a \(\pm\) button to assist in changing the number's sign after evaluation if you're directly entering it. Remember, negation is a fundamental operation, which essentially transforms positive results to negative, and vice-versa, reflecting a number's position relative to zero on the number line.