Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the common logarithm of each number to the nearest hundredth. $$ 3.75 $$

Step-by-Step Solution

Answer

The common logarithm of 3.75 is approximately 0.57.

Step 1: Understand the Problem

The problem asks for the common logarithm of the number 3.75 to the nearest hundredth. The common logarithm is the logarithm to the base 10, denoted as \( \log_{10} \) or simply as \( \log \).

Step 2: Calculate the Logarithm

Use a calculator to find the common logarithm of 3.75. Enter \( \log(3.75) \) into your calculator. You should get a value that is approximately 0.5740312677.

Step 3: Round the Logarithm

The answer needs to be rounded to the nearest hundredth. The number 0.5740312677 rounded to the nearest hundredth is 0.57 because the third decimal place (4) is less than 5, so we do not round up.

Key Concepts

Understanding Common Logarithms

Common logarithms are a type of logarithm that uses base 10. They are incredibly useful in many areas such as scientific calculations and can simplify complex multiplication or division operations by converting them into addition and subtraction of exponents. A common logarithm of a number, say 100, is the power to which 10 must be raised to equal 100, which is 2 because \(10^2 = 100\). Common logarithms are usually written as \(\log \), without specifying the base, implying a default base of 10. They are especially handy in contexts involving powers of ten, like scientific notation, as the base 10 aligns perfectly with this system.

Exploring Base 10 Logarithms

The base 10 logarithm, often referred to as the common logarithm, simplifies understanding powers and exponents. The notation \(\log_{10} \) is frequently abbreviated to \(\log\) because base 10 is so widely used. When you compute the base 10 logarithm of a number, you're essentially finding out 'how many 10s do you multiply together to get that number'. For instance, finding the \( \log_{10} 1000 \) is equivalent to finding what power of 10 equals 1000. Since 1000 is \(10^3\), \( \log_{10} 1000 = 3\). Utilizing base 10 logarithms simplifies complex number manipulations by reducing multiplication to addition. This is why they are prevalent in fields involving large computations.

Rounding Numbers: Basics and Importance

Rounding numbers plays a crucial role in ensuring that numerical results remain manageable and easy to interpret. When we talk about rounding to the nearest hundredth, we're focusing on the second decimal place. This type of rounding is particularly important when precision is necessary but storing or working with long decimal numbers becomes impractical. To round a number to the nearest hundredth, consider the digit in the third decimal place.

• If it is 5 or more, increase the second decimal by one.
• If it is less than 5, leave the second decimal unchanged.

Applying this rule, the number 0.574 when rounded to the nearest hundredth becomes 0.57 as the third place digit (4) is less than 5. Rounding ensures simplicity and clarity, making calculations more straightforward and less prone to errors while maintaining acceptable accuracy.