Problem 22 - Algebra for College Students

Question

Evaluate each logarithmic expression. \(\log _{3} 9\)

Step-by-Step Solution

Answer

The value of \(\log_{3} 9\) is 2.

Step 1: Understand What Logarithms Represent

The expression \(\log_{3} 9\) asks us to find the power to which the base 3 must be raised to produce the number 9. Mathematically, it asks us to solve \(3^x = 9\).

Step 2: Rewrite 9 as a Power of 3

Recognize that 9 is a power of 3. Specifically, 9 can be rewritten as \(3^2\). This helps us match the base 3 to the number 9, simplifying the equation.

Step 3: Solve for the Exponent

Since \(9 = 3^2\), we equate the exponents in the equation \(3^x = 3^2\) so that \(x = 2\). This shows that the expression \(\log_{3} 9 = 2\).

Key Concepts

Exponents

An exponent indicates how many times a number, known as the base, is multiplied by itself. It is a powerful mathematical notation that simplifies the expression of large numbers or repeated multiplication. For example, when we have the expression \(3^2\), the number 3 is multiplied by itself 2 times, resulting in 9.

• The exponent is the small number placed at the upper right of the base.
• If there is no exponent written, it implies the exponent is 1. For example, \(3 = 3^1\).
• A power of zero, such as \(3^0\), always equals 1, since any number to the power of zero is equal to 1.

Understanding exponents provides the foundation for comprehending logarithms, as logarithms are essentially dealing with powers in reverse.

Base and Power

In mathematics, the base is the number that is being repeatedly multiplied, and the power is the exponent that tells us how many times to multiply the base. In the context of logarithms, the base becomes crucial, as it helps to determine the value of the logarithmic expression.

• The base is written in normal-sized numbers.
• The power or exponent is written smaller, to notify of its operation on the base.

When you are solving a problem like \(3^x = 9\), you are using a base of 3 raised to an unknown power \(x\). The process involves understanding how to find a missing power by rewriting numbers with the same base, which simplifies calculations.

Logarithmic Expressions

Logarithmic expressions involve finding the exponent or power that a base must be raised to reach a specific number. They make solving for exponents more structured and provide a systematic way to approach exponential equations.

• A logarithm such as \(\log_{3} 9\) asks you directly: “To what power must we raise 3 to get 9?”
• The solution to the expression \(\log_{3} 9\) is the exponent found, which is 2, because \(3^2 = 9\).
• Logarithms can convert multiplication into addition, division into subtraction, powers into products, and roots into divisions, thus simplifying complex calculations in mathematics.

Gaining familiarity with logarithmic expressions is beneficial for understanding exponential growth and decay in fields like science, finance, and statistics.