Problem 1 - Algebra for College Students

Question

Use a calculator to find each common logarithm. Express answers to four decimal places. $$ \log 7.24 $$

Step-by-Step Solution

Answer

The common logarithm of 7.24 is approximately 0.8595.

Step 1: Understand the Task

You are asked to find the common logarithm, which is the logarithm to the base 10, of the number 7.24. This needs to be expressed to four decimal places.

Step 2: Input the Number

Enter the number 7.24 into your calculator. Most scientific calculators have a 'log' button, which is used for computing common logarithms.

Step 3: Calculate the Logarithm

Press the 'log' button on your calculator once the number 7.24 is entered. The calculator will compute the logarithm of 7.24 to the base 10.

Step 4: Record the Answer

The calculator will display a decimal number. This result represents the logarithm of 7.24 to base 10. Round this number to four decimal places for your final answer.

Key Concepts

Base 10 Logarithm

The base 10 logarithm, often referred to as the common logarithm, is a concept that simplifies working with exponential relationships by transforming them into linear ones. When you see \(\log\), it usually means you're dealing with a base 10 logarithm unless otherwise specified. The expression \(\log_{10}\) (or just \(\log\)) calculates the power to which 10 must be raised to produce a given number.

In the original exercise, we are asked to find \(\log 7.24\), which means, "What power must 10 be raised to in order to get 7.24?" This transformation is very useful in science and engineering, making it easier to handle large ranges of values.

• Calculates powers of 10
• Simplifies operations involving large numbers
• Common in scientific fields

Scientific Calculator

A scientific calculator is an essential tool used for performing complex mathematical functions, including finding logarithms. Most scientific calculators have a 'log' button that you use for calculating the base 10 or common logarithm.

To solve the exercise, you would enter 7.24 into the calculator and simply press the 'log' button. The device will automatically perform the necessary calculation of \(\log_{10} 7.24\).

• Look for the 'log' button
• Key in the number first, then press 'log'
• Provides an accurate and quick result

These tools are designed to be user-friendly to help you easily solve logarithmic, exponential, and trigonometric problems without error.

Decimal Places Accuracy

In mathematical calculations, precision in representation is key. This is where the concept of decimal places accuracy comes into play. Decimal places refer to the number of digits shown to the right of the decimal point in a number.

In the exercise, the answer is expressed with four decimal places, ensuring precision without unnecessary complexity. Rounding is often used here. After calculating the common logarithm with a scientific calculator, you might get more than four decimal places as your result. It is important to round your final result to four decimal places to meet the accuracy requirements of your exercise.

• Precision with decimal places stabilizes numerical results
• Four decimal places is standard for logarithmic calculations
• Rounding is essential when more precision than needed is given

Guidelines typically suggest rounding up if the next digit is 5 or greater, adding precision to your results to make them compatible with expected scientific norms.