Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the common logarithm of each number to the nearest hundredth. $$ 100 $$

Step-by-Step Solution

Answer

The common logarithm of 100 is 2.00.

Step 1: Understanding Logarithm Basics

The common logarithm is the logarithm with base 10, also denoted as \( \log_{10} \). So, when asked to find the common logarithm of a number, we are seeking \( \log_{10}(x) \).

Step 2: Apply the Logarithm Definition

The problem asks for \( \log_{10}(100) \). By the definition of logarithms, \( \log_{10}(x) = y \) is equivalent to saying \( 10^y = x \). We are seeking the value of \( y \) such that \( 10^y = 100 \).

Step 3: Solve the Logarithm Equation

We know that \( 10^2 = 100 \). Therefore, \( y = 2 \), meaning \( \log_{10}(100) = 2 \).

Step 4: Round to the Nearest Hundredth

Even though \( 2 \) is an exact answer, in this case, rounding to the nearest hundredth keeps the number the same: 2.00.

Key Concepts

Understanding Base 10 Logarithms

The base 10 logarithm, often referred to as the "common logarithm," uses 10 as its base. It is represented mathematically as \( \log_{10}(x) \). This means that when you are looking for the common logarithm of a number, you’re essentially solving for a power or exponent that 10 must be raised to in order to result in the original number you have. For example, in the problem given, where we need to find \( \log_{10}(100) \), we're trying to determine which power 10 needs to be raised to so that we get 100. Here, it turns out that 10 raised to the power of 2 equals 100. Therefore, \( \log_{10}(100) = 2 \). This idea of finding which power results in a specific number is the core of understanding what logarithms are about.

Exploring the Definition of Logarithms

Logarithms are the inverse operation of exponentiation, just like how subtraction is the inverse of addition. The logarithm of a number is the exponent by which the base must be raised to yield that number. In mathematical terms, \( \log_{b}(x) = y \) means \( b^y = x \). This concept allows us to solve expressions and equations involving powers in a more straightforward manner.

• If you are asked what \( \log_{10}(100) \) is, you look for the number \( y \) such that \( 10^y = 100 \).
• If you can determine that 10 squared (i.e., \( 10^2 \)) equals 100, then you have found that \( \log_{10}(100) \) is 2.

This understanding is crucial for solving problems involving exponential growth or decay, scientific notation, and many other mathematical and scientific computations.

Rounding Numbers: A Simple Guide

Rounding is a technique used to simplify numbers, making them easier to work with while maintaining approximate accuracy. For instance, when asked to round to the nearest hundredth, you focus on the second digit after the decimal point. Here's how you can round numbers:

• Identify the decimal place you are rounding to, which in this case is the hundredth position, or second place after the decimal.
• Look at the digit immediately after your place of interest.
• If this digit is 5 or higher, round the digit in the hundredth place up by one unit.
• If it's less than 5, leave the digit in the hundredth place as it is.

In our example, the logarithm of 100 is exactly 2, which means it remains 2 when rounded to the nearest hundredth, represented as 2.00. This process helps ensure clarity, especially when dealing with repeating decimals or lengthy computations.