Problem 2 - Intermediate Algebra

Question

For Problems \(1-10\), use a calculator to find each common logarithm. Express answers to four decimal places. \(\log 2.05\)

Step-by-Step Solution

Answer

\( \log 2.05 \approx 0.3118 \).

Step 1: Understanding the Task

We need to find the common logarithm, denoted as \( \log \), of the number 2.05. A common logarithm has a base of 10.

Step 2: Using the Calculator

Turn on your scientific calculator. Ensure it has a logarithm function, often labeled as \( \log \).

Step 3: Calculating the Logarithm

Input the number 2.05 into the calculator. Then, press the \( \log \) button to compute the common logarithm of 2.05.

Step 4: Reading the Result

The calculator displays a result for \( \log 2.05 \). Record this value, ensuring the answer is rounded to four decimal places.

Step 5: Rounding the Answer

If the calculator's display shows more than four decimal places, round the number appropriately. For example, if the result is 0.3010299957, it rounds to 0.3010 to four decimal places.

Key Concepts

Scientific Calculator Usage

A scientific calculator is an essential tool for tackling mathematical problems, especially those that involve logarithms. These calculators are equipped with functions that go beyond basic arithmetic operations. One of these advanced functions is the logarithm function, which typically calculates logarithms with base 10 by default, known as common logarithms.

To use the logarithm feature on a scientific calculator, locate the button labeled as "log". This button is specifically designed for computing common logarithms, which are logarithms with base 10.

• First, ensure your scientific calculator is turned on.
• Next, enter the number for which you wish to find the logarithm.
• Finally, press the "log" button.

Once these steps are followed, your calculator processes the command and presents the logarithm of the given number on its display. It's a straightforward process that greatly simplifies finding logarithmic values without needing to consult logarithm tables or perform manual calculations.

Decimal Places Rounding

When working with common logarithms on a scientific calculator, you often encounter results with many decimal places. Rounding these numbers to a specific number of decimal places is important for precision and clarity, particularly in academic and scientific settings.

The concept of rounding involves adjusting the digits of a number to maintain its value as close as possible to the original, while keeping it to the desired number of decimal places. This exercise specifically asks for the result to be rounded to four decimal places. Here's how you can do it:

• Identify the number at the fourth decimal place.
• Check the number immediately following it (the fifth decimal place) to determine if you need to round up or stay.
• If the fifth decimal place is 5 or greater, increase the fourth decimal place by one. If it is less than 5, leave the fourth decimal place unchanged.

For instance, if your calculator displays a result of 0.3010299957 for the logarithm, rounding it to four decimal places gives 0.3010. This ensures the representation of the number is precise and manageable for practical use.

Logarithm Base 10

The common logarithm, often referred to simply as "log", utilizes base 10. This is the standard log base used in many scientific and mathematical calculations.

Logarithms answer the question, "To what power must the base be raised, to produce a certain number?" For common logarithms, you’re finding what power 10 must be raised to, to equal the given number.

• The notation "\( \log 2.05 \)" means you are finding the power that 10 must be raised to get 2.05.
• It’s denoted \( \log_{10} \, 2.05 \), but the base 10 is implied and thus not always written in the notation.
• This kind of logarithm is very practical and commonly used in fields ranging from astronomy to economics.

Understanding and using logarithms base 10 allows for easier computation and comprehension of various complex multipliers, especially in scientific calculations and data analysis.