Problem 45 - Algebra A Combined Function

Question

Solve. \(\log _{3} 9=x\)

Step-by-Step Solution

Answer

\( x = 2 \)

Step 1: Identify the Logarithmic Equation

The given problem is \( \log_{3} 9 = x \), which asks for the power \( x \) that 3 must be raised to in order to obtain 9.

Step 2: Convert to Exponential Form

Using the property of logarithms that states \( \log_{b} a = c \) is equivalent to \( b^c = a \), rewrite the equation as an exponential equation: \( 3^x = 9 \).

Step 3: Express 9 as a Power of 3

Recognize that 9 can be expressed as \( 3^2 \). Thus, substitute it into the equation: \( 3^x = 3^2 \).

Step 4: Equate the Exponents

Since the bases are the same (both are 3), equate the exponents: \( x = 2 \).

Step 5: Verify the Solution

Substitute \( x = 2 \) back into the original logarithmic equation to verify: \( \log_{3} 9 = 2 \) because \( 3^2 = 9 \). Therefore, \( x = 2 \) is correct.

Key Concepts

Properties of Logarithms

Logarithms are mathematical operations that help us solve equations involving exponential relationships. Understanding their properties is crucial in interpreting and solving logarithmic equations.
At its core, a logarithm asks a question like: "To what power should a certain number (the base) be raised to obtain another number?" For example, in the exercise \( \log_{3} 9 = x \), it asks "to what power should 3 be raised to result in 9?" This property is known as the definition of a logarithm: \( \log_{b} a = c \) where \( b^c = a \).

• The **Product Rule** states that \( \log_{b} (MN) = \log_{b} M + \log_{b} N \).
• The **Quotient Rule** states that \( \log_{b} \left(\frac{M}{N}\right) = \log_{b} M - \log_{b} N \).
• The **Power Rule** states that \( \log_{b} (M^n) = n \cdot \log_{b} M \).

These rules allow us to break down more complex logarithmic expressions into simpler parts. Recognizing these properties aids in converting and simplifying expressions, crucial for solving and understanding logarithmic equations.

Exponential Equations

Exponential equations are an integral part of understanding logarithms as they represent the inverse operation. When we have an equation in logarithmic form like \( \log_{b} a = c \), its exponential form is \( b^c = a \).
This transition from logarithmic to exponential form is vital. It helps simplify the problem by comparing or solving for unknown exponents. When dealing with numbers like in the example \( 3^x = 9 \), it's usually clear to re-frame them as powers of the same base if possible, here \( 9 = 3^2 \).
Solving the equation \( 3^x = 3^2 \) is straightforward now because the exponents must be equal if the bases are the same, leading to \( x = 2 \).
Understanding and converting equations from logarithmic to exponential form empowers you to tackle more complex problems systematically by equating powers of similar bases.

Verifying Logarithmic Solutions

Once a potential solution is found in a logarithmic equation, verification is crucial to ensure accuracy. This step confirms that the solution fits the original condition of the problem.
In our case, after identifying \( x = 2 \) as a solution, it is important to plug it back into the original equation \( \log_{3} 9 = x \). This verification involves checking if \( 3^2 \) indeed equals 9, which it does.
Verification reinforces confidence in the solution and ensures that no steps were skipped or erroneously computed. It is an essential part of problem solving, especially in math, where small errors can lead to incorrect conclusions.
Through verification, we confirm that our understanding of the problem, the solution process, and the final answer are all aligned correctly. It is a step that should never be overlooked.