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Chapter 12: Problem 28

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

Short Answer

Expert verified
x = 2

Step by step solution

01

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: \[ 3^x = 3^2 \]
02

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: \[ x = 2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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