What Is The Solution To This Equation? 9 X − 1 = 2 9^x - 1 = 2 9 X − 1 = 2 A. − 1 2 -\frac{1}{2} − 2 1 ​ B. 1C. 2D. 1 2 \frac{1}{2} 2 1 ​

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic and exponential functions. In this article, we will focus on solving the equation $9^x - 1 = 2$, which is a classic example of an exponential equation. We will break down the solution step by step, using various mathematical techniques to arrive at the final answer.

Understanding Exponential Equations

Exponential equations are equations that involve exponential functions, which are functions of the form $f(x) = a^x$, where $a$ is a positive real number. Exponential equations can be linear or non-linear, and they can involve various mathematical operations, such as addition, subtraction, multiplication, and division.

The Equation $9^x - 1 = 2$

The equation $9^x - 1 = 2$ is a non-linear exponential equation, which involves an exponential function with base $9$. To solve this equation, we need to isolate the exponential term and then use various mathematical techniques to find the value of $x$.

Step 1: Isolate the Exponential Term

The first step in solving the equation $9^x - 1 = 2$ is to isolate the exponential term. We can do this by adding $1$ to both sides of the equation, which gives us:

$$9^x = 3$$

Step 2: Use Logarithms to Solve for $x$

The next step in solving the equation $9^x = 3$ is to use logarithms to find the value of $x$. We can use the logarithmic function to rewrite the equation as:

$$x = \log_9(3)$$

Step 3: Evaluate the Logarithmic Expression

To evaluate the logarithmic expression $\log_9(3)$, we need to use the change of base formula, which states that:

$$\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$$

where $c$ is any positive real number. In this case, we can choose $c = 10$, which gives us:

$$\log_9(3) = \frac{\log_{10}(3)}{\log_{10}(9)}$$

Step 4: Simplify the Expression

To simplify the expression $\frac{\log_{10}(3)}{\log_{10}(9)}$, we need to use the fact that $\log_{10}(9) = 2\log_{10}(3)$. This gives us:

$$\frac{\log_{10}(3)}{\log_{10}(9)} = \frac{\log_{10}(3)}{2\log_{10}(3)} = \frac{1}{2}$$

Conclusion

In conclusion, the solution to the equation $9^x - 1 = 2$ is $x = \frac{1}{2}$. This is a classic example of an exponential equation, and it requires a deep understanding of algebraic and exponential functions to solve. By breaking down the solution step by step, we can arrive at the final answer and gain a deeper understanding of the underlying mathematical concepts.

Answer

The correct answer is:

• D. $\frac{1}{2}$

Discussion

This equation is a classic example of an exponential equation, and it requires a deep understanding of algebraic and exponential functions to solve. The solution involves isolating the exponential term, using logarithms to solve for $x$, and evaluating the logarithmic expression. By breaking down the solution step by step, we can arrive at the final answer and gain a deeper understanding of the underlying mathematical concepts.

Related Topics

• Exponential functions
• Logarithmic functions
• Algebraic equations
• Mathematical techniques for solving exponential equations

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Logarithmic Functions" by Math Open Reference
• [3] "Algebraic Equations" by Math Open Reference

Additional Resources

• Khan Academy: Exponential Functions
• Khan Academy: Logarithmic Functions
• MIT OpenCourseWare: Algebra and Calculus
  Frequently Asked Questions: Exponential Equations =====================================================

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic and exponential functions. In this article, we will answer some of the most frequently asked questions about exponential equations, including how to solve them, what are the common mistakes to avoid, and how to apply them in real-world scenarios.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is a positive real number.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponential term and then use various mathematical techniques to find the value of $x$. This can involve using logarithms, algebraic manipulations, and other mathematical techniques.

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

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

• Not isolating the exponential term
• Not using the correct mathematical techniques to solve the equation
• Not checking the domain of the exponential function
• Not considering the possibility of multiple solutions

Q: How do I apply exponential equations in real-world scenarios?

A: Exponential equations have many real-world applications, including:

• Modeling population growth and decline
• Modeling financial growth and decline
• Modeling chemical reactions and decay
• Modeling electrical and electronic circuits

Q: What are some examples of exponential equations?

A: Some examples of exponential equations include:

• $2^x = 8$
• $3^x = 27$
• $4^x = 64$
• $9^x - 1 = 2$

Q: How do I use logarithms to solve exponential equations?

A: To use logarithms to solve exponential equations, you need to use the change of base formula, which states that:

$$\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$$

where $c$ is any positive real number.

Q: What are some common applications of exponential equations?

A: Some common applications of exponential equations include:

• Modeling population growth and decline
• Modeling financial growth and decline
• Modeling chemical reactions and decay
• Modeling electrical and electronic circuits

Q: How do I check the domain of an exponential function?

A: To check the domain of an exponential function, you need to ensure that the base of the exponential function is positive and that the exponent is a real number.

Q: What are some common mistakes to avoid when checking the domain of an exponential function?

A: Some common mistakes to avoid when checking the domain of an exponential function include:

• Not checking the base of the exponential function
• Not checking the exponent of the exponential function
• Not considering the possibility of multiple solutions

Conclusion

In conclusion, exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic and exponential functions. By answering some of the most frequently asked questions about exponential equations, we can gain a deeper understanding of the underlying mathematical concepts and how to apply them in real-world scenarios.

Additional Resources

• Khan Academy: Exponential Functions
• Khan Academy: Logarithmic Functions
• MIT OpenCourseWare: Algebra and Calculus
• Wolfram Alpha: Exponential Equations

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Logarithmic Functions" by Math Open Reference
• [3] "Algebraic Equations" by Math Open Reference