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Chapter 10: Problem 95

In the following exercises, solve each equation. $$ 2^{3 x-8}=16 $$

Short Answer

Expert verified
x = 4

Step by step solution

01

- Understanding the base

The equation given is in terms of base 2. Recognize that 16 can be written as a power of 2. Specifically, 16 = 2^4.
02

- Rewriting the equation

Rewrite the original equation by replacing 16 with 2^4. 2^{3x - 8} = 2^4.
03

- Setting the exponents equal

Since the bases are the same, set the exponents equal to each other. 3x - 8 = 4.
04

- Solving for x

Solve the equation for x by isolating x. 3x - 8 = 4Add 8 to both sides: 3x = 12Divide both sides by 3:x = 4.

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Key Concepts

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

Base Recognition
When solving exponential equations, recognizing the base is the first crucial step. In this exercise, the equation is given in base 2. You should check if the other number (in this case, 16) can be expressed with the same base. Here, 16 can indeed be written as a power of 2, specifically, 16 = 2^4.
This step is important because if both sides of the equation share the same base, it simplifies the process greatly. This makes it easier to compare the exponents later.
Exponent Rules
Exponential equations often use exponent rules to simplify the equation. In this exercise, once you rewrite 16 as 2^4, you notice both sides of the equation now have base 2: 2^{3x - 8} = 2^4.
According to the exponent rules, when the bases are the same, you can set the exponents equal to each other. This greatly simplifies the equation into something much easier to solve. Remember, always ensure your bases match before applying these rules, as this is crucial for the next steps.
Isolating Variables
After setting the exponents equal, the next step is isolating the variable. For our example, once we know that 2^{3x - 8} = 2^4 translates to 3x - 8 = 4, our goal is to isolate x.
To do so, we perform basic algebraic operations to move other terms away from x.
  • Add 8 to both sides: 3x - 8 + 8 = 4 + 8, which simplifies to 3x = 12.
  • Then, divide both sides by 3: 3x / 3 = 12 / 3, which gives x = 4.
By isolating the variable, we find the solution in a systematic way.
Simplifying Equations
Simplifying equations helps in solving them efficiently. In this case, once you've rewritten the bases and set the exponents equal, you'll end up with a simpler equation. From 3x - 8 = 4, simple steps like adding 8 and dividing by 3, guided us to find x = 4.
Simplification reduces the complexity of the equation and makes it easier to see the solution path. Always aim to make the equation less complicated, step by step, as it makes solving the equation almost intuitive.

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Most popular questions from this chapter

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places. \(\left(\frac{1}{2}\right)^{x}=6\)

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