Problem 3 - Algebra for College Students

Question

Use a calculator to find each common logarithm. Express answers to four decimal places. $$ \log 52.23 $$

Step-by-Step Solution

Answer

\( \log 52.23 \approx 1.7177 \)

Step 1: Understand the Problem

The exercise asks us to find the common logarithm (base 10) of the number 52.23 and express it to four decimal places. The common logarithm is often represented as \( \log_{10} \) or simply \( \log \).

Step 2: Use a Calculator

To solve, enter 52.23 into the calculator and use the logarithm function (often labeled as 'log'). Make sure the calculator is set to compute base 10 logarithms, which is standard in most calculators.

Step 3: Interpret the Calculator's Answer

After pressing the 'log' button, the calculator will display the value of \( \log 52.23 \). Note this value down.

Step 4: Round to Four Decimal Places

If the calculator displays more than four decimal places, round the result to the fourth decimal place. Ensure proper rounding rules are applied: if the fifth decimal is 5 or greater, round the fourth decimal up; otherwise, leave it as is.

Key Concepts

Base 10 Logarithms

Common logarithms, also known as base 10 logarithms, are a way to represent exponential growth in a linear form. When you see \( \log x \), this is shorthand for saying "the power to which the base 10 must be raised to get \( x \)." This implies that the common logarithm of 100 is 2 because \( 10^2 = 100 \). The concept of logarithms is essential in simplifying calculations, especially in fields like science and engineering, where very large or small numbers are frequent. To further understand, consider \( 10^3 = 1000 \), thus \( \log 1000 = 3 \). Logarithms allow us to work with the exponent (3), which is usually a more manageable number compared to 1000. Always remember, the base here is 10, so make sure when calculating that the method or calculator is set for common logs.

Using a Calculator for Logarithms

Calculators have revolutionized the way we compute common logarithms. Most scientific calculators have a dedicated 'log' button for this purpose. To find \( \log 52.23 \), simply key in 52.23, then press the 'log' button. It's important to confirm that your calculator is set to base 10 mode, which is usually the default setting for the 'log' button. Some calculators might need you to press 'log' before entering the number, so it's wise to refer to your calculator's manual if you're unsure.
Using a calculator not only speeds up the process but also ensures precision, which is crucial when dealing with precise measurements or scientific data.

Rounding Decimal Places

Rounding decimal places is a common practice in mathematics to make numbers easier to work with while maintaining a fair level of accuracy. When you get a result from a calculator and it’s longer than you need, you will likely need to round it.
For instance, if you are asked to give a number rounded to four decimal places and your calculator shows 1.717862, you'll look at the fifth decimal place to decide how to round.

• If the fifth decimal place is 5 or greater, increase the fourth decimal place by 1.
• If the fifth decimal is less than 5, keep the fourth decimal as it is.

In our example, since the fifth place is 8, we round up to 1.7179. This practice helps in reporting results concisely, maintaining a standard and understood accuracy level across various problems.