Which Graph Shows The Solution To The Equation 4 X − 3 = 8 4^{x-3} = 8 4 X − 3 = 8 ?

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Introduction

In this article, we will explore the solution to the equation $4^{x-3} = 8$ and determine which graph represents the solution. We will use algebraic manipulation and graphing techniques to find the correct graph.

Understanding the Equation

The given equation is $4^{x-3} = 8$. To solve this equation, we need to isolate the variable $x$. We can start by rewriting the equation in a more manageable form.

Rewriting the Equation

We can rewrite the equation as follows:

$$4^{x-3} = 8$$

$$\Rightarrow 4^{x-3} = 2^3$$

$$\Rightarrow (2^2)^{x-3} = 2^3$$

$$\Rightarrow 2^{2(x-3)} = 2^3$$

Now, we can equate the exponents:

$$2(x-3) = 3$$

$$\Rightarrow x-3 = \frac{3}{2}$$

$$\Rightarrow x = \frac{3}{2} + 3$$

$$\Rightarrow x = \frac{9}{2}$$

Graphing the Solution

To graph the solution, we need to find the value of $x$ that satisfies the equation. We have already found that $x = \frac{9}{2}$.

Graph 1

The first graph is a logarithmic graph with a base of 2. The graph has a vertical asymptote at $x = 3$.

import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(-10, 10, 400)
y = np.log2(x)
plt.plot(x, y)
plt.axvline(x=3, color='r', linestyle='--')
plt.show()

Graph 2

The second graph is an exponential graph with a base of 2. The graph has a horizontal asymptote at $y = 0$.

import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(-10, 10, 400)
y = 2**x
plt.plot(x, y)
plt.axhline(y=0, color='r', linestyle='--')
plt.show()

Graph 3

The third graph is a logarithmic graph with a base of 4. The graph has a vertical asymptote at $x = 3$.

import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(-10, 10, 400)
y = np.log2(x/2)
plt.plot(x, y)
plt.axvline(x=3, color='r', linestyle='--')
plt.show()

Graph 4

The fourth graph is an exponential graph with a base of 4. The graph has a horizontal asymptote at $y = 0$.

import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(-10, 10, 400)
y = 4**x
plt.plot(x, y)
plt.axhline(y=0, color='r', linestyle='--')
plt.show()

Conclusion

Based on the algebraic manipulation and graphing techniques used, we can conclude that the correct graph is the third graph, which is a logarithmic graph with a base of 4.

Why is the Third Graph the Correct Graph?

The third graph is the correct graph because it represents the solution to the equation $4^{x-3} = 8$. The graph has a vertical asymptote at $x = 3$, which corresponds to the value of $x$ that satisfies the equation. The graph also has a logarithmic shape, which is consistent with the equation.

What is the Significance of the Third Graph?

The third graph is significant because it represents the solution to a real-world problem. The equation $4^{x-3} = 8$ can be used to model a variety of real-world situations, such as population growth or chemical reactions. The graph provides a visual representation of the solution, which can be used to make predictions and decisions.

What are the Implications of the Third Graph?

The third graph has several implications. Firstly, it shows that the solution to the equation is a logarithmic function. This has implications for the behavior of the function, such as its rate of growth and its asymptotic behavior. Secondly, it shows that the vertical asymptote at $x = 3$ is a critical point in the function. This has implications for the behavior of the function near this point, such as its rate of change and its limit as $x$ approaches 3.

What are the Limitations of the Third Graph?

The third graph has several limitations. Firstly, it only represents the solution to the equation $4^{x-3} = 8$. It does not provide information about the behavior of the function for other values of $x$. Secondly, it assumes that the function is continuous and differentiable. This may not be the case in all real-world situations. Finally, it assumes that the graph is a perfect representation of the function. In reality, the graph may be subject to errors and approximations.

What are the Future Directions of the Third Graph?

The third graph has several future directions. Firstly, it can be used to model a variety of real-world situations, such as population growth or chemical reactions. Secondly, it can be used to make predictions and decisions based on the behavior of the function. Finally, it can be used to develop new mathematical models and techniques for solving equations.

Conclusion

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Introduction

In our previous article, we explored the solution to the equation $4^{x-3} = 8$ and determined that the third graph, a logarithmic graph with a base of 4, represents the solution. In this article, we will answer some frequently asked questions about the equation and its solution.

Q: What is the equation $4^{x-3} = 8$?

A: The equation $4^{x-3} = 8$ is an exponential equation that can be rewritten as $2^{2(x-3)} = 2^3$. This equation represents a relationship between the variable $x$ and the constant 8.

Q: How do I solve the equation $4^{x-3} = 8$?

A: To solve the equation $4^{x-3} = 8$, we can use algebraic manipulation to isolate the variable $x$. We can start by rewriting the equation as $2^{2(x-3)} = 2^3$. Then, we can equate the exponents and solve for $x$.

Q: What is the solution to the equation $4^{x-3} = 8$?

A: The solution to the equation $4^{x-3} = 8$ is $x = \frac{9}{2}$.

Q: Which graph represents the solution to the equation $4^{x-3} = 8$?

A: The third graph, a logarithmic graph with a base of 4, represents the solution to the equation $4^{x-3} = 8$.

Q: What is the significance of the third graph?

A: The third graph is significant because it represents the solution to a real-world problem. The equation $4^{x-3} = 8$ can be used to model a variety of real-world situations, such as population growth or chemical reactions.

Q: What are the implications of the third graph?

A: The third graph has several implications. Firstly, it shows that the solution to the equation is a logarithmic function. This has implications for the behavior of the function, such as its rate of growth and its asymptotic behavior. Secondly, it shows that the vertical asymptote at $x = 3$ is a critical point in the function.

Q: What are the limitations of the third graph?

A: The third graph has several limitations. Firstly, it only represents the solution to the equation $4^{x-3} = 8$. It does not provide information about the behavior of the function for other values of $x$. Secondly, it assumes that the function is continuous and differentiable. This may not be the case in all real-world situations.

Q: What are the future directions of the third graph?

A: The third graph has several future directions. Firstly, it can be used to model a variety of real-world situations, such as population growth or chemical reactions. Secondly, it can be used to make predictions and decisions based on the behavior of the function. Finally, it can be used to develop new mathematical models and techniques for solving equations.

Q: How can I use the third graph in real-world applications?

A: The third graph can be used in a variety of real-world applications, such as:

• Modeling population growth or decline
• Modeling chemical reactions
• Making predictions and decisions based on the behavior of the function
• Developing new mathematical models and techniques for solving equations

Conclusion

In conclusion, the third graph represents the solution to the equation $4^{x-3} = 8$. It is a logarithmic graph with a base of 4 and has several implications and limitations. The graph can be used in a variety of real-world applications, such as modeling population growth or chemical reactions.

Additional Resources

For more information on the equation $4^{x-3} = 8$ and its solution, please refer to the following resources:

• Mathematical Modeling
• Exponential Functions
• Logarithmic Functions

References

• [1] "Mathematical Modeling" by Wikipedia
• [2] "Exponential Functions" by Wikipedia
• [3] "Logarithmic Functions" by Wikipedia