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Chapter 4: Problem 22

Solve the following equations for \(x\). $$2^{4-x}=8$$

Short Answer

Expert verified
x = 1

Step by step solution

01

Rewrite 8 as a power of 2

Recognize that 8 can be rewritten as a power of 2. Since 8 = 2^3, the equation becomes: 2^{4-x} = 2^3.
02

Equate the exponents

Since the bases are the same, set the exponents equal to each other: 4 - x = 3.
03

Solve for x

Subtract 4 from both sides to isolate -x: -x = 3 - 4 Simplify the right-hand side: -x = -1 Multiply both sides by -1 to solve for x: x = 1.

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Key Concepts

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

Exponents
Exponents are a way to express repeated multiplication of a number by itself. For example, in the term 2^3, the number 2 is the base, and 3 is the exponent, indicating that 2 should be multiplied by itself three times: 2 * 2 * 2. Understanding how to manipulate and rewrite terms involving exponents is key to solving exponential equations.
In our example, we started with the equation 2^{4-x} = 8. The first step was to express 8 as a power of 2 because it's easier to deal with equations when both sides have the same base. We recognized that 8 is equal to 2^3.
Equating Exponents
In exponential equations, if the bases are the same on both sides of the equation, we can set the exponents equal to each other. This is because for two expressions with the same base to be equal, their exponents must also be equal.
So, in our transformed equation by converting 8 to 2^3, we have 2^{4-x} = 2^3. Now that the bases on both sides are identical, we focus on equating the exponents directly: 4 - x = 3.
Isolation of Variables
To solve for the variable, we need to isolate it. This means rearranging the equation so that the variable is on one side and the constants are on the other. Here, we start with the equation: 4 - x = 3.
  • Subtract 4 from both sides:

    • - x = 3 - 4
  • Simplify the right-hand side:

    • - x = -1
  • Multiply both sides by -1 to solve for x:

    • x = 1
So, the value of x that satisfies the equation is 1. Isolating variables is a crucial step in solving equations because it allows us to determine the value of unknown variables effectively.

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Most popular questions from this chapter

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