Problem 21 - 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. \(x=\frac{7}{8}\)

Step-by-Step Solution

Answer

The logarithm of \(x = \frac{7}{8}\) is approximately -0.133 when rounded to three decimal places.

Step 1: Input Value into Function

Input \(x = \frac{7}{8}\) into the function \(f(x) = \log{x}\). To enter the fractional value into a scientific calculator, use the division symbol to divide 7 by 8.

Step 2: Evaluate the Logarithm

Evaluate the logarithm of \(\frac{7}{8}\) using your scientific calculator. If your calculator has a dedicated natural logarithm (ln) button, use that button. Otherwise, use the generic logarithm button (log) and then divide the result by log(e) where 'e' is Euler’s number (approximately equal to 2.718). Note that log operations are often not performed on fractions directly, so the calculated result is the logarithm of a positive number less than 1. Make sure your calculator is in the correct mode!

Step 3: Round the Answer

Round your answer to three decimal places. Your calculator should provide enough decimal places. You only need to keep the first three decimal places, and consider the fourth place to decide whether to round up or down the third decimal place.

Key Concepts

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions might seem challenging at first, but with the right tools and understanding, it can become straightforward. When you encounter a problem like evaluating the logarithm of \( x = \frac{7}{8} \), the first step is to understand what you're calculating. The function \( f(x) = \log x \) typically denotes the common logarithm, which is the logarithm base 10, unless stated otherwise. In this context, you are finding what power you need to raise 10 to get \( \frac{7}{8} \).
Understanding this means knowing that:

• Logarithms are the inverse operation to exponentiation.
• Common logarithms use base 10.
• The argument \( x \) in \( \log x \) must be positive.

The approach for evaluating \( \log(\frac{7}{8}) \) involves using a calculator to input the fraction directly by dividing 7 by 8 to find its decimal equivalent. This allows the logarithmic function to correctly interpret and compute it.

Rounding Decimal Places

Rounding decimal places is an essential skill when you're dealing with precision in mathematical computations. After calculating \( \log(\frac{7}{8}) \), you'll receive a number with potentially many decimal places. To round this to three decimal places, follow these steps:

• Identify the first three digits after the decimal point.
• Look at the fourth digit to decide if you should round up or remain the same.
• Increase the third digit by one if the fourth digit is 5 or greater.
• Leave the third digit as is if the fourth digit is less than 5.

Rounding to three decimal places means you are ensuring that your result is precise enough for most mathematical applications without being overly detailed. It helps in making calculations simpler and more intuitive to understand.

Using a Calculator for Logarithms

Using a calculator for logarithms takes some understanding of your calculator's functions. On most scientific calculators, you will find a button labeled "LOG" or "log" for base 10 logarithm computations. Here's how to use it to calculate \( \log(\frac{7}{8}) \):

• Enter the fraction using the division operation, i.e., enter "7 ÷ 8" to see the decimal 0.875.
• With 0.875 displayed, press the "log" button.
• Your result, which is \( \log(0.875) \), will be displayed.
• If you have a natural logarithm ("ln") button, ensure you are not using it, as it operates with base \( e \), not 10.

Make sure your calculator is set correctly and you understand the type of logarithm it's calculating. Using the calculator efficiently ensures the right result and enhances your problem-solving abilities in logarithmic evaluations.