Select The Correct Answer.What Is The Solution To This Equation?${ 9^z - 1 = 2 }$A. 2 B. 1 C. { -\frac{1}{2}$}$ D. { \frac{1}{2}$}$

Introduction

In mathematics, equations are a fundamental concept that help us understand and describe various phenomena. Solving equations is a crucial skill that requires a deep understanding of mathematical concepts and techniques. In this article, we will focus on solving a specific equation, $9^z - 1 = 2$, and explore the different approaches to find the solution.

Understanding the Equation

The given equation is $9^z - 1 = 2$. To solve this equation, we need to isolate the variable $z$. The equation can be rewritten as $9^z = 3$. This is a simple exponential equation, and we can use various techniques to solve it.

Using Algebraic Manipulation

One approach to solve the equation is to use algebraic manipulation. We can start by adding 1 to both sides of the equation, which gives us $9^z = 3 + 1$. This simplifies to $9^z = 4$. Now, we can take the logarithm of both sides to solve for $z$. Taking the logarithm of both sides gives us $z \log 9 = \log 4$. We can then divide both sides by $\log 9$ to get $z = \frac{\log 4}{\log 9}$.

Using Exponential Properties

Another approach to solve the equation is to use exponential properties. We can rewrite the equation as $9^z = 3$. Since $9 = 3^2$, we can rewrite the equation as $(3^2)^z = 3$. Using the property of exponents, we can simplify this to $3^{2z} = 3$. Now, we can equate the exponents to get $2z = 1$. Dividing both sides by 2 gives us $z = \frac{1}{2}$.

Using Logarithmic Properties

We can also use logarithmic properties to solve the equation. Taking the logarithm of both sides gives us $\log 9^z = \log 3$. Using the property of logarithms, we can rewrite this as $z \log 9 = \log 3$. Now, we can divide both sides by $\log 9$ to get $z = \frac{\log 3}{\log 9}$.

Evaluating the Options

Now that we have solved the equation, we can evaluate the options. The correct solution is $z = \frac{1}{2}$. Let's compare this with the options:

• A. 2: This is not the correct solution.
• B. 1: This is not the correct solution.
• C. $-\frac{1}{2}$: This is not the correct solution.
• D. $\frac{1}{2}$: This is the correct solution.

Conclusion

In this article, we have solved the equation $9^z - 1 = 2$ using various techniques, including algebraic manipulation, exponential properties, and logarithmic properties. We have also evaluated the options and found that the correct solution is $z = \frac{1}{2}$. This demonstrates the importance of understanding mathematical concepts and techniques in solving equations.

Final Answer

Q: What is the equation $9^z - 1 = 2$ trying to solve?

A: The equation $9^z - 1 = 2$ is trying to solve for the value of $z$ that satisfies the equation.

Q: What is the first step in solving the equation $9^z - 1 = 2$?

A: The first step in solving the equation $9^z - 1 = 2$ is to isolate the variable $z$. We can do this by adding 1 to both sides of the equation, which gives us $9^z = 3$.

Q: How do we solve the equation $9^z = 3$?

A: We can solve the equation $9^z = 3$ by using various techniques, including algebraic manipulation, exponential properties, and logarithmic properties.

Q: What is the correct solution to the equation $9^z - 1 = 2$?

A: The correct solution to the equation $9^z - 1 = 2$ is $z = \frac{1}{2}$.

Q: Why is $z = \frac{1}{2}$ the correct solution?

A: $z = \frac{1}{2}$ is the correct solution because it satisfies the equation $9^z - 1 = 2$. When we substitute $z = \frac{1}{2}$ into the equation, we get $9^{\frac{1}{2}} - 1 = 2$, which is true.

Q: What are some common mistakes to avoid when solving the equation $9^z - 1 = 2$?

A: Some common mistakes to avoid when solving the equation $9^z - 1 = 2$ include:

• Not isolating the variable $z$ correctly
• Not using the correct techniques to solve the equation
• Not checking the solution to make sure it satisfies the equation

Q: How can I practice solving equations like $9^z - 1 = 2$?

A: You can practice solving equations like $9^z - 1 = 2$ by working through practice problems and exercises. You can also try solving different types of equations, such as linear equations and quadratic equations.

Q: What are some real-world applications of solving equations like $9^z - 1 = 2$?

A: Solving equations like $9^z - 1 = 2$ has many real-world applications, including:

• Modeling population growth and decay
• Analyzing financial data and making predictions
• Solving optimization problems in business and economics

Conclusion

In this article, we have answered some frequently asked questions about solving the equation $9^z - 1 = 2$. We have covered topics such as the first step in solving the equation, the correct solution, and common mistakes to avoid. We have also discussed real-world applications of solving equations like $9^z - 1 = 2$.