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Chapter 4: Problem 22

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

Short Answer

Expert verified
The value of x is 1.

Step by step solution

01

Recognize the Powers of 2

Both 2 and 8 can be expressed as powers of 2. Notice that 8 is the same as 2^3.
02

Rewrite the Equation

Rewrite the equation with both sides as powers of 2. The equation becomes: \[ 2^{4-x} = 2^3 \]
03

Set the Exponents Equal

Since the bases are the same, you can set the exponents equal to each other: \[ 4 - x = 3 \]
04

Solve for x

Solve the equation for x by isolating x: \[ 4 - x = 3 \] Subtract 3 from both sides: \[ 4 - 3 = x \] Thus, \[ x = 1 \]

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Key Concepts

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

Powers of 2
In mathematics, powers of 2 are numbers in the form of 2 to the power of an integer (exponent). For example,
  • 2^1 = 2,
  • 2^2 = 4,
  • 2^3 = 8,
  • 2^4 = 16,
and so on. If you have an equation where you need to solve for a variable, expressing numbers as powers of 2 can make things simpler. For example, in the given exercise, recognizing that 8 can be written as 2^3 helps us to rewrite the equation more efficiently. It also allows us to use properties of exponents.
Rewriting Equations
Rewriting equations is a crucial step in solving exponential equations. It involves transforming the given equation into a more manageable form. For our exercise, we start with: 2^{4-x} = 8. Since 8 can be written as 2^3, we rewrite the equation as: 2^{4-x} = 2^3.This step allows us to work with the bases more easily, making it easier to proceed with the solution. The goal is to have the equation in a form where both sides have the same base.
Isolating Variables
Isolating the variable means getting the variable by itself on one side of the equation. This often involves a series of algebraic manipulations. In our exercise, once we have 2^{4-x} = 2^3, we can use the property of exponents that if the bases are equal, the exponents must be equal. This gives us: 4 - x = 3.Next, we need to isolate x. To do this, we solve for x in the equation: 4 - x = 3. Subtracting 3 from both sides, we get: 4 - 3 = x,which simplifies to: x = 1.Now, the variable x is isolated.
Setting Exponents Equal
Setting exponents equal means using the property that if two exponential expressions with the same base are equal, their exponents must be equal too. In our exercise, once we rewrote 2^{4-x} = 8 as 2^{4-x} = 2^3, we can set the exponents equal to each other: 4 - x = 3.This is due to the fact that the only way for 2^{a} = 2^{b} is if a = b.By setting the exponents equal, we transition from an exponential equation to a simple linear equation which is easier to solve.

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