Problem 15 - Algebra A Combined Function

Question

Find the exact value of each logarithm. $$ \log 100 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We are tasked with finding the exact value of the logarithmic expression \( \log 100 \). This logarithm is assumed to be base 10, also known as the common logarithm.

Step 2: Identify the Base and Argument

In this expression, \( \log 100 \), the base is 10 (common logarithm), and the argument is 100. We need to determine what power 10 must be raised to in order to produce 100.

Step 3: Express 100 as a Power of 10

The number 100 can be expressed as a power of 10: \( 100 = 10^2 \).

Step 4: Apply the Definition of Logarithms

According to the definition of logarithms, \( \log_b x = y \) means \( b^y = x \). For \( \log_{10} 100 \), we have \( 10^y = 100 \). Since \( 10^2 = 100 \), it follows that \( y = 2 \).

Step 5: Write the Final Answer

Thus, the exact value of \( \log 100 \) is 2.

Key Concepts

Common Logarithm

A common logarithm is simply a logarithm that has a base of 10. In mathematics, common logarithms are often used for ease of calculations and are represented as \( \log \). When you see a logarithm written without a base specified, you can safely assume it is a base 10, or common, logarithm. For instance, when you see \( \log 100 \), it is shorthand for \( \log_{10} 100 \).

This notation is primarily used because base 10 is intrinsically linked to our number system, which is decimal or base 10. The common logarithm is incredibly convenient for both calculations and scientific notation, providing an easier way to handle large numbers and understand exponential relationships.

• Common logarithms simplify the representation of large values.
• Used in scientific notation to easily convey really big or really small numbers.

Base 10

The base 10 system is integral to understanding common logarithms. In the expression \( \log_{10} 100 \), the number 10 is the base of the logarithm, meaning it’s the number that is raised to a certain power to result in the number 100 (the argument).

Base 10 can be seen everywhere in our daily lives because we use a decimal system. This is the numbering system based on powers of ten, which is why understanding base 10 is essential when learning about logarithms.

• The base of the logarithm dictates the exponential relationship.
• Our decimal system naturally uses base 10.
• Base 10 logarithms help simplify complex calculations.

Power of 10

In logarithms, expressing a number as a power of 10 is crucial for finding solutions. For example, to solve \( \log 100 \), you first express 100 as \( 10^2 \) since 100 is equivalent to 10 raised to the power of 2.

This step is important because the logarithm of a number is essentially asking, "What power do we raise the base (10) to, in order to get this number?" Understanding this interpretation of powers of 10 will make working with logarithms much more intuitive and straightforward.

• Reveals the relationship between exponential expressions and logarithms.
• Necessary for translating between exponential and logarithmic forms.

Definition of Logarithms

Logarithms are mathematical operations that help us solve equations involving exponents. The definition of a logarithm \( \log_b x = y \) means that \( b^y = x \). So for \( \log_{10} 100 \), it is essentially finding the number \( y \) such that \( 10^y = 100 \).

Applying the definition of logarithms involves translating an exponential equation into logarithmic form to find unknowns, such as powers. It helps simplify otherwise complex multiplication and division problems by turning them into addition and subtraction.

• Logarithms are the inverse operations of exponentials.
• Facilitates solving equations where the unknown is an exponent.
• Provides a framework for understanding exponential growth and decay.