Problem 28 - Algebra A Combined Function

Question

By inspection, find the value for \(x\) that makes each statement true. \(3^{x}=9\)

Step-by-Step Solution

Answer

x = 2

Step 1: Express Both Sides with the Same Base

Notice both 3 and 9 are powers of 3. Write 9 as a power of 3: \[ 9 = 3^2 \] Now the equation becomes: \[ 3^x = 3^2 \]

Step 2: Equate the Exponents

Since the bases on both sides of the equation are the same, the exponents must be equal for the statement to be true: \[ x = 2 \]

Key Concepts

Exponents

Exponents are a fundamental concept in Algebra. They are a shorthand way to express repeated multiplication. For instance, in the expression \( a^n \), the number \( a \) is known as the base, and \( n \) is the exponent. This expression signifies that the base \( a \) is multiplied by itself \( n-1 \) additional times.

• An exponent helps in simplifying expressions by avoiding repeated multiplication. For example, \( 2^3 \) is a compact way of writing \( 2 \times 2 \times 2 \).
• The exponent can be an integer, positive, negative, or zero, each with specific rules.

In the exercise given, understanding exponents allows us to manipulate the equation \( 3^x = 9 \) by re-expressing 9 using 3 as a base.

Equations

An equation is a statement that asserts the equality of two expressions. Equations are like balanced scales, where both sides represent the same value.

• Equations typically contain one or more variables, like \( x \), that need to be solved.
• The goal is usually to find the value of the unknown variable that makes the equation true.

In the given problem, we have an equation with an exponential expression, \( 3^x = 9 \), meaning we need to determine the value of \( x \) that makes the left-hand side equal to the right-hand side. This involves using properties of exponents and equations.

Powers

The term "powers" is closely related to exponents. When you write \( a^n \), you are actually writing a "power" of \( a \).

• The expression \( a^n \) can be read as "\( a \) to the power of \( n \)."
• "Power" refers to the entire expression that involves both the base and the exponent.

Powers simplify the process of expressing large numbers. They are critical in solving exponential equations like \( 3^x = 9 \) because they allow us to reframe numbers as powers of a common base, thereby making comparison and calculation simpler.

Base Conversion

Base conversion involves expressing a number in terms of another base, especially when dealing with exponents. It's essential for solving equations like \( 3^x = 9 \), where you need to express both numbers using a common base.

• In our case, 9 is rewritten as \( 3^2 \) to match the base of 3 on the left-hand side of the equation.
• This process simplifies the equation to \( 3^x = 3^2 \), enabling a straightforward comparison of the exponents.

Base conversion is a useful skill in algebra when handling exponential relationships, making the solutions more accessible and easier to understand.