Problem 89 - Elementary and Intermediate Algebra

Question

Use a calculator to find each value. Give answers to four decimal places. See Using Your Calculator: Evaluating Logarithms. $$ \log 0.00467 $$

Step-by-Step Solution

Answer

The value of \\( \log 0.00467 \\) is approximately -2.3315.

Step 1: Understanding the Problem

We need to find the approximate value of the logarithm of the number 0.00467 to four decimal places using a calculator. Logarithms help us determine the power to which a base number must be raised to get another number.

Step 2: Input into Calculator

Make sure your calculator is set to base 10, which is the default setting for most everyday log calculations. Enter the number 0.00467 into the calculator and press the 'log' button to compute the logarithm.

Step 3: Reading the Calculator Result

After pressing the 'log' button, the calculator displays a result. This is often a negative number because 0.00467 is less than 1, assuming a base 10 logarithm.

Step 4: Rounding to Four Decimal Places

Take the calculator's result and round it to four decimal places, ensuring precision as required by the problem. For instance, if the calculator shows -2.331482, round this to -2.3315.

Key Concepts

base 10 logarithms

Logarithms are a fascinating and powerful concept in mathematics. They represent the power to which a specified base number must be raised to produce a given number. The base 10 logarithm, often seen in the form \(\log_{10}(x)\), simplifies to just \(\log(x)\) because base 10 is the default for common logarithms. This makes it one of the most used types of logarithms in daily calculations and sciences.
If you encounter a decimal number like 0.00467, you're essentially finding out what exponent 10 must be raised to in order to result in 0.00467. This helps transform multiplication into addition, which simplifies complex calculations.
Understanding this can empower you to use logarithms effectively, especially when dealing with large or small numbers.

rounding numbers

Rounding numbers is a crucial skill for ensuring precision, especially when dealing with decimals. It allows us to simplify numbers while maintaining their significant figures as required by the exercise or context.
When rounding to four decimal places, focus on the fifth decimal place. Here's a quick guide:

• If the fifth digit is 5 or more, increase the fourth decimal place by one. Example: \(-2.331482\) rounds to \(-2.3315\).
• If the fifth digit is less than 5, simply keep the fourth digit as it is. Example: \(-2.334462\) rounds to \(-2.3344\).

Being adept at rounding not only keeps your work neat but also aligns your answers with standardized expectations.

calculator usage

Calculators are indispensable tools in the modern world of mathematics and science. To effectively use them for evaluating logarithms, you should be aware of several key points. Here are the basic steps:

• Ensure your calculator is set to base 10, which is common for general logarithmic calculations.
• Input the number in question (for example, 0.00467) and press the 'log' button. This gives you the logarithm of the number with base 10.
• Review the result carefully. For small numbers less than 1, the result will typically be a negative number, indicating that a small decimal number requires a negative power of 10.

This process is made easy with calculators, but an understanding of how the tool operates enhances your confidence in both basic and advanced applications.