The Solution Set Of $2^{x+1}=8$ Is:A. ${2}$ B. $ { 3 } \{3\} { 3 } [/tex] C. ${4}$ D. ${5}$

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will focus on the solution set of the exponential equation $2^{x+1}=8$. This equation involves a base of 2 and an exponent of $x+1$, and we need to find the value of $x$ that satisfies the equation. We will analyze the given options and determine the correct solution set.

Understanding Exponential Equations

Exponential equations are of the form $a^x=b$, where $a$ is the base and $b$ is the result. In this case, the base is 2, and the result is 8. To solve the equation, we need to isolate the variable $x$. We can start by rewriting the equation as $2^{x+1}=23$.

Properties of Exponents

When dealing with exponential equations, it's essential to understand the properties of exponents. One of the key properties is the power rule, which states that $a^{m+n}=a^ma^n$. We can use this property to simplify the equation and isolate the variable $x$.

Simplifying the Equation

Using the power rule, we can rewrite the equation as $2^{x+1}=23$. Since the bases are the same, we can equate the exponents, which gives us $x+1=3$.

Solving for x

Now that we have the equation $x+1=3$, we can solve for $x$ by subtracting 1 from both sides. This gives us $x=2$.

Analyzing the Solution Set

The solution set of an equation is the set of all possible values of the variable that satisfy the equation. In this case, we have found that $x=2$ is the solution to the equation $2^{x+1}=8$. However, we need to determine if this is the only solution or if there are other values of $x$ that also satisfy the equation.

Checking the Options

Let's analyze the given options and determine if they match the solution we found.

Option A: {2}

This option suggests that the solution set is $\{2\}$. Since we found that $x=2$ is the solution to the equation, this option seems plausible.

Option B: {3}

This option suggests that the solution set is $\{3\}$. However, we found that $x=2$ is the solution, not $x=3$.

Option C: {4}

This option suggests that the solution set is $\{4\}$. However, we found that $x=2$ is the solution, not $x=4$.

Option D: {5}

This option suggests that the solution set is $\{5\}$. However, we found that $x=2$ is the solution, not $x=5$.

Conclusion

Based on our analysis, we can conclude that the correct solution set is $\{2\}$. This means that the only value of $x$ that satisfies the equation $2^{x+1}=8$ is $x=2$.

Final Answer

The final answer is $\boxed{A}$.

Frequently Asked Questions

Q: What is the solution set of the equation $2^{x+1}=8$?

A: The solution set is $\{2\}$.

Q: How do I solve exponential equations?

A: To solve exponential equations, you need to understand the properties of exponents and use techniques such as the power rule to simplify the equation and isolate the variable.

Q: What is the power rule of exponents?

A: The power rule of exponents states that $a^{m+n}=a^ma^n$.

References

• [1] "Exponential Equations" by Math Open Reference
• [2] "Properties of Exponents" by Khan Academy
• [3] "Solving Exponential Equations" by Purplemath

Related Topics

• [1] "Linear Equations"
• [2] "Quadratic Equations"
• [3] "Polynomial Equations"

Tags

• Exponential Equations
• Properties of Exponents
• Power Rule
• Solution Set
• Linear Equations
• Quadratic Equations
• Polynomial Equations
  

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will provide a comprehensive guide to exponential equations, including a Q&A section that addresses common questions and concerns.

What are Exponential Equations?

Exponential equations are of the form $a^x=b$, where $a$ is the base and $b$ is the result. In this case, the base is 2, and the result is 8. To solve the equation, we need to isolate the variable $x$.

How Do I Solve Exponential Equations?

To solve exponential equations, you need to understand the properties of exponents and use techniques such as the power rule to simplify the equation and isolate the variable. Here are some steps to follow:

1. Understand the properties of exponents: Exponents have several properties, including the power rule, which states that $a^{m+n}=a^ma^n$.
2. Simplify the equation: Use the power rule to simplify the equation and isolate the variable.
3. Equating the exponents: Since the bases are the same, we can equate the exponents, which gives us $x+1=3$.
4. Solving for x: Now that we have the equation $x+1=3$, we can solve for $x$ by subtracting 1 from both sides.

What is the Power Rule of Exponents?

The power rule of exponents states that $a^{m+n}=a^ma^n$. This rule allows us to simplify exponential expressions by combining the exponents.

How Do I Use the Power Rule to Solve Exponential Equations?

To use the power rule to solve exponential equations, follow these steps:

1. Identify the base and the exponent: Identify the base and the exponent in the equation.
2. Apply the power rule: Use the power rule to simplify the equation and isolate the variable.
3. Equating the exponents: Since the bases are the same, we can equate the exponents, which gives us $x+1=3$.
4. Solving for x: Now that we have the equation $x+1=3$, we can solve for $x$ by subtracting 1 from both sides.

What is the Solution Set of an Exponential Equation?

The solution set of an equation is the set of all possible values of the variable that satisfy the equation. In this case, we have found that $x=2$ is the solution to the equation $2^{x+1}=8$.

Q&A

Q: What is the solution set of the equation $2^{x+1}=8$?

A: The solution set is $\{2\}$.

Q: How do I solve exponential equations?

A: To solve exponential equations, you need to understand the properties of exponents and use techniques such as the power rule to simplify the equation and isolate the variable.

Q: What is the power rule of exponents?

A: The power rule of exponents states that $a^{m+n}=a^ma^n$.

Q: How do I use the power rule to solve exponential equations?

A: To use the power rule to solve exponential equations, follow these steps:

1. Identify the base and the exponent: Identify the base and the exponent in the equation.
2. Apply the power rule: Use the power rule to simplify the equation and isolate the variable.
3. Equating the exponents: Since the bases are the same, we can equate the exponents, which gives us $x+1=3$.
4. Solving for x: Now that we have the equation $x+1=3$, we can solve for $x$ by subtracting 1 from both sides.

Q: What is the solution set of an exponential equation?

A: The solution set of an equation is the set of all possible values of the variable that satisfy the equation.

Q: How do I check if an equation is an exponential equation?

A: To check if an equation is an exponential equation, look for the base and the exponent. If the equation has a base and an exponent, it is an exponential equation.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

• Not understanding the properties of exponents: Make sure you understand the properties of exponents, including the power rule.
• Not simplifying the equation: Make sure you simplify the equation using the power rule.
• Not equating the exponents: Make sure you equate the exponents since the bases are the same.
• Not solving for x: Make sure you solve for x by subtracting 1 from both sides.

Conclusion

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. By following the steps outlined in this article, you can solve exponential equations and understand the properties of exponents. Remember to check your work and avoid common mistakes.

Final Answer

The final answer is $\boxed{A}$.

Frequently Asked Questions

Q: What is the solution set of the equation $2^{x+1}=8$?

A: The solution set is $\{2\}$.

Q: How do I solve exponential equations?

A: To solve exponential equations, you need to understand the properties of exponents and use techniques such as the power rule to simplify the equation and isolate the variable.

Q: What is the power rule of exponents?

A: The power rule of exponents states that $a^{m+n}=a^ma^n$.

References

• [1] "Exponential Equations" by Math Open Reference
• [2] "Properties of Exponents" by Khan Academy
• [3] "Solving Exponential Equations" by Purplemath

Related Topics

• [1] "Linear Equations"
• [2] "Quadratic Equations"
• [3] "Polynomial Equations"

Tags

• Exponential Equations
• Properties of Exponents
• Power Rule
• Solution Set
• Linear Equations
• Quadratic Equations
• Polynomial Equations