Problem 52 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to evaluate the logarithm. Round your result to three decimal places.\(\ln 36.7\)

Step-by-Step Solution

Answer

The rounded natural logarithm of 36.7 is 3.600. Before rounding, the calculator may give a longer decimal.

Step 1: Access a Scientific Calculator

Firstly, we need to access a scientific calculator. These calculators can be physical devices, or they may be digital programs or apps on your phone or computer. Basic calculators won't work here because we need a calculator that has a button for 'ln'

Step 2: Input the Value

Next, you need to input the value you want to take the log of into the calculator. In this exercise, the value is 36.7.

Step 3: Compute the Natural Logarithm

Find the button labeled 'ln' on your calculator. Press this button - your calculator should take the natural logarithm of the number you entered. Note: On some calculators, you may need to press the 'ln' button before you enter the number.

Step 4: Round the Result

Lastly, we must round our result to three decimal places as the exercise instructs.

Key Concepts

Scientific Calculator Usage

When it comes to mathematical computations involving exponents and logarithms, a scientific calculator is your indispensable tool. Unlike basic calculators, scientific calculators come with a plethora of functions, including the ability to calculate the natural logarithm, denoted as 'ln'.

To start, locate the 'ln' button on your calculator. This special key directly computes the natural logarithm of a number. The process may vary slightly depending on your calculator's model. In some cases, you will enter the number first and then press the 'ln' button. In other models, you must press 'ln' first, followed by the number. After pressing the necessary keys, the calculator displays the logarithmic value. Always refer to your calculator’s manual for precise instructions on its use.

Remember the most important steps:

• Turn on the calculator and ensure it's in the correct mode for performing logarithmic calculations.
• Enter the number or press 'ln', depending on your calculator model.
• Press 'ln' or enter the number, again depending on the model.
• Read the result displayed on the screen.

With these steps, utilizing a scientific calculator to find logarithms becomes a simple task.

Natural Logarithm Calculation

The natural logarithm is a fundamental concept in mathematics, notably used in calculus and physics. Natural logarithms have the constant 'e' (approximately 2.71828) as their base. The natural logarithm of a number, 'x', written as \( \ln(x) \), tells us the power to which 'e' must be raised to obtain 'x'.

For example, if you wanted to find the natural logarithm of 36.7, you would be seeking the exponent that 'e' must be raised to in order to get 36.7. Calculating this without a calculator is complex and impractical, which is why we use calculators equipped with a 'ln' function. After entering the desired value and pressing the appropriate buttons, as mentioned in the 'Scientific Calculator Usage' section, the calculator will provide you with \( \ln(36.7) \), which is the power of 'e' needed to reach the number 36.7. This calculation is widely used in science and engineering fields to deal with exponential growth or decay, among other applications.

Rounding Numbers

In mathematics, rounding numbers is a technique used to reduce the digits in a number while keeping its value close to the original. Rounding is vital when precision is less critical, or when a more straightforward, less cumbersome number is preferred. The exercise specifies rounding to three decimal places. This means we want three digits to the right of the decimal point.

Here's how to round:

• Identify the fourth decimal place.
• If this digit is five or higher, increase the third decimal place by one.
• If it is four or lower, keep the third decimal place as is.
• Eliminate all digits beyond the third decimal place.

For instance, if the calculator displays 3.14159, we round this to 3.142 when rounding to three decimal places. Mastering this skill is essential, as rounding is frequently used in reporting measurements, financial transactions, and statistical data, where manageable numbers are often more useful than exact ones.