Problem 2 - Algebra for College Students

Question

Use a calculator to find each common logarithm. Express answers to four decimal places. $$ \log 2.05 $$

Step-by-Step Solution

Answer

The common logarithm of 2.05 is 0.3118.

Step 1: Understand the Common Logarithm

The common logarithm is the logarithm to base 10. When the exercise asks for \( \log 2.05 \), it is requesting the logarithm base 10 of 2.05.

Step 2: Use Calculator to Compute the Logarithm

Enter 2.05 into your calculator and press the 'log' button to find \( \log 2.05 \). Most scientific calculators have a dedicated button for common logarithms.

Step 3: Record the Answer

Ensure your calculator is set to display results to four decimal places. After computing, \( \log 2.05 \) returns approximately 0.3118.

Key Concepts

Logarithm Base 10

The logarithm base 10, often referred to as a common logarithm, is a convenient way of expressing how many times 10 must be multiplied by itself to reach a specific number. In mathematics, this is written using the expression \( \log_{10}(x) \) or simply \( \log(x) \). For instance, if you encounter \( \log 2.05 \), it means you're determining the power to which 10 must be raised to result in 2.05.
Common logarithms are especially prevalent in science and engineering because they simplify complex multiplications into simple additions. If you're dealing with large numbers or tiny fractions, the common logarithm compresses this complexity, making it easier to manage.
To visualize: if \( \log 100 = 2 \), it means 10 multiplied by itself 2 times equals 100. Understanding this concept is essential for solving logarithmic exercises, such as the one asking you to evaluate \( \log 2.05 \).

Use of Calculator in Math

Calculators are invaluable tools in modern mathematics, especially when dealing with complex functions like logarithms. By using a scientific calculator, you can easily compute the logarithm base 10, such as \( \log 2.05 \).
Here's how you can efficiently use the calculator for finding common logarithms:

• Ensure your calculator is in standard mode, ready for basic operations.
• Input the number for which you need the logarithm. For our exercise, this is 2.05.
• Locate the 'log' button. It's typically labeled clearly on the calculator.
• Press the 'log' button after entering the number.

In a matter of seconds, your calculator will display the answer. This ease of use highlights the calculator's role in streamlining mathematical computations, allowing for quick and precise calculations without needing manual logarithmic tables or lengthy paper-and-pencil methods.

Decimal Places in Calculations

Understanding how to round and display results accurately is crucial to math exercises like calculating logarithms. Decimal places refer to the number of digits shown after the decimal point in a number. In many exercises, you're asked to express the answer to a certain number of decimal places for precision.
In our logarithmic example, we're tasked to express \( \log 2.05 \) to four decimal places. This means rounding the result to the fourth digit after the decimal point. When rounding:

• Identify the fourth decimal digit following the decimal point in your answer.
• Check the digit right after the fourth one: if it's 5 or greater, round up the fourth digit by one. Otherwise, keep it as is.

For example, if the calculator shows 0.31184874, you round it to 0.3118. Applying the correct number of decimal places ensures consistency and accuracy in calculations, which is vital for clear communication of mathematical results.