Problem 20 - Intermediate Algebra

Question

In this Study Set, assume that all variables represent positive numbers and \(b \neq 1\). Evaluate each expression. See Example \(1 .\) $$ \log _{9} 9 $$

Step-by-Step Solution

Answer

\( \log_9 9 = 1 \)

Step 1: Understanding Logarithms

The logarithmic expression \( \log_b a \) asks for the power \( x \) that the base \( b \) must be raised to in order to result in \( a \). In this problem, we need to evaluate \( \log_9 9 \).

Step 2: Identifying Base and Result

The expression \( \log_9 9 \) indicates that the base is 9 and the result we want is also 9. We are trying to find what power \( 9^x \) equals 9.

Step 3: Recognizing the Power of 1

We know that any number raised to the power of 1 is equal to the number itself, here \( 9^1 = 9 \). Thus, the power that 9 must be raised to obtain 9 is 1.

Step 4: Writing the Solution

Since \( 9^1 = 9 \), we can conclude that \( \log_9 9 = 1 \). Therefore, the expression evaluates to 1.

Key Concepts

Logarithmic Functions

Logarithmic functions are a fascinating and essential part of mathematics. They help us understand how many times we need to multiply a base number in order to achieve another number. Here's how they work:

• The notation \( \log_b a \) means "the logarithm of \( a \) with base \( b \)."
• This expression answers the question: "To what power must \( b \) be raised to result in \( a \)?"
• Logarithmic functions are the inverse of exponential functions. Where exponential functions raise a base to a power, logarithms take the result and tell you the power.

Understanding logarithmic functions is crucial, as they are widely used in various fields like science, engineering, and even finance. They're a fundamental tool for dealing with exponential growth, decay, and much more.

Base and Exponent

The concept of base and exponent is central to understanding logarithmic expressions:

• The **base** in \( \log_b a \) refers to the number that is being multiplied. It's like the starting block for any logarithmic problem.
• The **exponent** is the power to which the base is raised. It's what you solve for in logarithmic expressions.

In the problem \( \log_9 9 \), both the base and the number we want are the same, making it simple:

• Base: 9
• Exponent: 1, because \( 9^1 = 9 \)

Knowing how to identify the base and search for the right exponent answers offers a clear path through more complex logarithmic problems, and understanding these components is a vital skill in math.

Evaluating Expressions

Evaluating logarithmic expressions can be straightforward once you understand the mechanics. Let’s break down the process using \( \log_9 9 \):

• Recognize the expression format: \( \log_b a \), where you determine what power of \( b \) results in \( a \).
• Identify that base and number are identical, simplifying our problem as \( b = a \).
• Remember that any number to the power of 1 is itself: \( 9^1 = 9 \).

Thus, when base and result are the same, the logarithm will always equal 1 because the base only needs itself raised to the first power to equal itself. Evaluating such expressions requires practice but becomes easier with understanding these foundational concepts.