Problem 81 - College Algebra

Question

Evaluate or simplify each expression without using a calculator. $$ \log 100 $$

Step-by-Step Solution

Answer

The value of \( \log 100 \) is 2.

Step 1: Identify Base and Number

In the given expression \( \log 100 \), we identify that the base of the log isn't written. It means that by default it is 10 and the number is 100.

Step 2: Apply Logarithm Definition

Applying the definition of logarithms, \( \log 100 \) is the number which raised to base 10 equals 100. We know that \( 10^2 = 100 \). Therefore, \( \log 100 = 2 \) because \( 10^2 = 100 \)

Key Concepts

Logarithm Definition

A logarithm is a way to express the relationship between a base (a positive number) and a number through an exponent. Specifically, the logarithm of a number is the exponent by which the base must be raised to produce that number. For example, if you see the expression \( \log_b x \), it is asking, 'To what exponent must we raise the base \( b \) to get \( x \)?'

To make this more concrete, let's return to our exercise example, \( \log 100 \). This is commonly written without a base, which means by convention it is to base 10 (we'll cover this more in the Base of Logarithm section). The question it poses is, 'What power do we raise 10 to get 100?' The solution, as seen in the exercise, is 2, because \( 10^2 = 100 \).

Base of Logarithm

The base of a logarithm is crucial because it dictates how the number is broken down exponentially. In a logarithm expression \( \log_b x \), the 'b' represents the base. If no base is specified, it is assumed to be 10, an agreed-upon standard known as the common logarithm. Other frequent bases include 'e' (an irrational number approximately equal to 2.718), known as the natural logarithm and written as \( \ln x \), and base 2, used frequently in computer science.

In our exercise, the absence of a base implies that it is a common logarithm with base 10. Recognizing the implied base is an important step toward solving logarithmic problems, as it guides how to rewrite the expression in exponential form.

Exponential Form

Understanding exponential form is essential when working with logarithms. An expression in exponential form has a base raised to a power or exponent, and it represents repeated multiplication. The notation \( b^n \) tells us that the base \( b \) is multiplied by itself \( n \) times.

When you convert a logarithmic expression to exponential form, you're finding a more familiar way to express the same equation. For example, the logarithmic equation \( \log_b x = n \) can be rewritten in exponential form as \( b^n = x \). This is the technique used in the exercise solution where the logarithm \( \log 100 \) is expressed as the exponential equation \( 10^2 = 100 \), thus clearly showing that the value of the logarithm is 2.