Problem 8 - Intermediate Algebra

Question

For Problems \(1-10\), use a calculator to find each common logarithm. Express answers to four decimal places. \(\log 0.04376\)

Step-by-Step Solution

Answer

\( \log(0.04376) \approx -1.3592 \)

Step 1: Understanding Common Logarithms

Common logarithms are logarithms with base 10. When you see a logarithm written as \( \log x \), it is assumed to have base 10. Therefore, we are interested in finding \( \log_{10}(0.04376) \).

Step 2: Using a Calculator

Turn on your calculator and ensure it is set to use base 10, if required (most calculators automatically deal with base 10 for \( \log \) function). Enter the number, 0.04376, into your calculator, and then press the \( \log \) button.

Step 3: Recording the Result

Once you press the \( \log \) button, the calculator should display the value of \( \log(0.04376) \). Record this value ensuring it is correct to four decimal places.

Step 4: Verify the Precision

Double-check the decimal value to ensure it is correct to four decimal places. If rounding is necessary, ensure you round correctly based on the numbers in the fifth decimal place.

Key Concepts

Base 10 Logarithms

In mathematics, common logarithms are logarithms with a base of 10. If you see a logarithm written simply as \( \log x \), it is assumed to have base 10. These types of logarithms are frequently used in science and engineering due to their simplicity and the prevalence of the decimal number system.
In essence, a base 10 logarithm tells you the power or exponent to which the number 10 must be raised to produce a given number. For example, \( \log 1000 = 3 \) because \( 10^3 = 1000 \).
This logarithm system is particularly useful when dealing with extremely large or small numbers, helping to scale values to more manageable ranges.

Calculator Usage in Mathematics

Calculators are invaluable tools for performing mathematical operations, including finding logarithms. Most modern scientific calculators are equipped to compute base 10 logarithms with a dedicated \( \log \) button.

• Before using a calculator to find logarithms, make sure it is powered on and ready for logarithmic functions. While most calculators automatically use base 10 for the log function, it's always a good practice to check.
• To calculate \( \log 0.04376 \) using a calculator, simply input the number and press the \( \log \) key. The display will show the result directly.

Having a reliable calculator can save time and reduce the likelihood of errors when manually computing complex operations, especially when dealing with non-integer values.

Rounding and Precision

When dealing with logarithms or any mathematical computations, the precision of your answer is crucial. In this exercise, ensuring the logarithm is expressed to four decimal places is essential for accuracy.
Here's how to ensure proper rounding and precision:

• Look at the fifth decimal place to decide whether to round up or down. If this digit is 5 or more, round up the fourth decimal. Otherwise, keep the fourth decimal the same.
• Recording your result accurately is important, especially in scientific fields where even slight variations can alter outcomes.

By mastering rounding techniques, you can ensure that your answers are as precise and correct as possible, which is vital in various real-world applications.