Problem 24 - Precalculus with Limits

Question

Use a calculator to evaluate \(f(x)=\log x\) at the indicated value of \(x .\) Round your result to three decimal places. \(x=96.75\)

Step-by-Step Solution

Answer

The value of the log of \(96.75\) rounded to three decimal places is 1.986. Note: The precise result may vary slightly depending upon your calculator.

Step 1: Input the value of \(x\) into the calculator

In this step, the number \(96.75\) is inputted into a calculator to determine its logarithm. This can be done by finding the button or function on your calculator that denotes 'log' and typing in \(96.75\).

Step 2: Calculate the value

After inputting the value of \(x\) into your calculator, perform the calculation. The calculator will display a result.

Step 3: Round to three decimal places

Once you have the result, round it to three decimal places. If the fourth decimal is greater than or equal to 5, round up the third decimal. If the fourth decimal digit is less than 5, keep the third decimal as it is.

Key Concepts

Understanding Logarithmic Functions

Logarithmic functions are inverses of exponential functions and are fundamental in various fields such as mathematics, physics, and engineering. They allow us to solve equations where the variable is an exponent, which would be difficult or impossible to solve otherwise. A function of the form
\( f(x) = \log x \)
is the base-10 logarithm, which answers the question: 'To what power must 10 be raised, to yield the number x?'. For example, when evaluating \( \log 100 \) the answer is 2, because \( 10^2 = 100 \). These functions have unique properties, such as \( \log(ab) = \log a + \log b \) and \( \log(a/b) = \log a - \log b \), which are useful in manipulating and solving logarithmic equations.

Using Scientific Calculators for Logarithms

Scientific calculators are equipped to handle a variety of functions, including logarithmic calculations. To evaluate a logarithm like \( \log 96.75 \) using a scientific calculator, follow these steps:

• Locate the button labeled 'log', which represents the base-10 logarithm.
• Enter the value you wish to find the logarithm of. Here, you would input 96.75.
• Press 'Enter' or the '=' button to display the logarithmic value on-screen.

Calculators simplify the otherwise complex operations involved in evaluating logarithms, especially for numbers that do not have a base-10 integer as their logarithm. Remember, different calculators may have slightly different steps or button layouts, so it's essential to be familiar with your device.

Rounding Decimals in Logarithmic Calculations

Rounding is a mathematical technique used to reduce the number of decimal places in a number to make it easier to work with. When rounding to three decimal places, you look at the fourth decimal place. If it is 5 or more, you increase the third decimal place by one, known as rounding up. If it is less than 5, you leave the third decimal place as it is, which is rounding down.
For example, if the calculator shows a result for \( \log 96.75 \) as 1.98573, to round to three decimal places, observe the fourth decimal place (7 in this case), which means you round up. The rounded answer would be 1.986. It's crucial to follow these rounding rules accurately, as rounding can affect the precision of your calculations, and in fields such as science and finance, precision is often key.