Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the common logarithm of each number to the nearest hundredth. $$ 8.56 $$

Step-by-Step Solution

Answer

The common logarithm of 8.56 is approximately 0.93.

Step 1: Understanding the Common Logarithm

The common logarithm is the logarithm to base 10, often denoted as \( \log_{10} \), and it can be written as \( \log \) when the base is understood as 10. We are tasked to find \( \log_{10}(8.56) \) to the nearest hundredth.

Step 2: Calculating the Common Logarithm

Using a calculator, or logarithm table, calculate \( \log_{10}(8.56) \). Input the value 8.56 to find the logarithm.

Step 3: Rounding the Result

The calculated value from the calculator is approximately \( 0.932383 \). We need to round this to the nearest hundredth, which is two decimal places.

Step 4: Finishing the Rounding

Look at the third decimal place (\(3\)) to decide if we round up or keep the second decimal place as is. Since 3 is less than 5, we do not round up. Therefore, the result remains \(0.93\).

Key Concepts

Logarithm Base 10

The concept of a logarithm base 10 is straightforward to grasp. It's often simply called the common logarithm and is used extensively in mathematics and science. When you see \( \log \), it usually means the common logarithm, with the base being understood as 10. Essentially, the logarithm of a number to a particular base is the power to which the base must be raised to produce that number.

For example, if you want to find \( \log_{10}(8.56) \), you're looking for the power to which 10 must be raised to get 8.56. Written out, it solves the equation:

• \( 10^x = 8.56 \)

Learning to understand logarithms allows you to analyze exponential growth or decay in real-world scenarios, like phone batteries charging or water purification through distillation. This foundational knowledge will make many mathematical topics more accessible.

Rounding Decimals

Rounding decimals is an important skill to master when working with mathematical results. When asked to round to the nearest hundredth, you are dealing with two decimal places. Let's say your calculated value is \(0.932383\).

Here's a simple way to approach rounding:

• Identify the decimal place you need to round to, in this case, the second decimal place.
• Look at the number immediately after it, which is the third decimal place.
• If this third decimal is 5 or higher, you round up. If it is less than 5, you round down.

In this example, the third decimal place is 3. Since 3 is less than 5, you round down, keeping the number as \(0.93\). Remembering these simple rules will help you handle rounding with ease.

Using a Calculator

Calculators are excellent tools for quickly finding logarithms, especially when dealing with numbers that are not as straightforward to estimate. Virtually any scientific calculator will have a \( \log \) button for calculating logarithms quickly and accurately. Here's a basic way to perform this function:

• Find the \( \log \) button, which stands for the logarithm base 10.
• Enter the number you want the logarithm of, such as 8.56.
• Press the \( \log \) button to get your result.

For our example, inputting 8.56 and pressing \( \log \) will yield approximately \(0.932383\). Calculators remove the tediousness of manual calculations, allowing you to focus on understanding and application. They prove invaluable for checking your work against manual solutions, offering quick reinforcement of learning concepts. Remember: calculators are there to assist, but understanding the why and how means you can work without them when needed.