Using Properties of Exponents to Solve the Equation

Oct 13, 2025 by recoveryshake.com 29 views

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Introduction

In mathematics, functions play a crucial role in describing relationships between variables. The functions $f$ and $g$ are given by $f(x)=4^{(5x-1)}$ and $g(x)=8^{(x/4)}$ . When solving the equation $f(x)=g(x)$ , the functions can be rewritten in equivalent forms so that the equation can be solved. In this article, we will explore the functions $f$ and $g$ , and discuss how they can be rewritten to facilitate the solution of the equation.

The Function $f(x)$

The function $f(x)$ is given by $f(x)=4^{(5x-1)}$ . This function can be rewritten in an equivalent form using the properties of exponents. Specifically, we can rewrite $4^{(5x-1)}$ as $(2^2)^{(5x-1)}$ . Using the property of exponents that states $(a^b)^c=a^{bc}$ , we can simplify this expression to $2^{2(5x-1)}$ . This is an equivalent form of the function $f(x)$ .

Simplifying the Function $f(x)$

We can further simplify the function $f(x)$ by evaluating the exponent. Specifically, we can rewrite $2^{2(5x-1)}$ as $2^{10x-2}$ . This is a simplified form of the function $f(x)$ .

The Function $g(x)$

The function $g(x)$ is given by $g(x)=8^{(x/4)}$ . This function can be rewritten in an equivalent form using the properties of exponents. Specifically, we can rewrite $8^{(x/4)}$ as $(2^3)^{(x/4)}$ . Using the property of exponents that states $(a^b)^c=a^{bc}$ , we can simplify this expression to $2^{3(x/4)}$ . This is an equivalent form of the function $g(x)$ .

Simplifying the Function $g(x)$

We can further simplify the function $g(x)$ by evaluating the exponent. Specifically, we can rewrite $2^{3(x/4)}$ as $2^{(3x)/4}$ . This is a simplified form of the function $g(x)$ .

Solving the Equation $f(x)=g(x)$

Now that we have rewritten the functions $f$ and $g$ in equivalent forms, we can solve the equation $f(x)=g(x)$ . Specifically, we can set the two functions equal to each other and solve for $x$ . This gives us the equation $2^{10x-2}=2^{(3x)/4}$ .

Using Properties of Exponents to Solve the Equation

We can use the properties of exponents to solve the equation $2^{10x-2}=2^{(3x)/4}$ . Specifically, we can equate the exponents and solve for $x$ . This gives us the equation $10x-2=(3x)/4$ .

Solving for $x$

We can solve for $x$ by isolating the variable on one side of the equation. Specifically, we can multiply both sides of the equation by 4 to eliminate the fraction. This gives us the equation $40x-8=3x$ . We can then add 8 to both sides of the equation to get $40x=3x+8$ . Finally, we can subtract $3x$ from both sides of the equation to get $37x=8$ . Dividing both sides of the equation by 37 gives us the solution $x=8/37$ .

Conclusion

In this article, we explored the functions $f$ and $g$ , and discussed how they can be rewritten in equivalent forms to facilitate the solution of the equation $f(x)=g(x)$ . We showed that the functions $f$ and $g$ can be rewritten as $f(x)=2^{10x-2}$ and $g(x)=2^{(3x)/4}$ , respectively. We then used the properties of exponents to solve the equation $f(x)=g(x)$ , and found the solution to be $x=8/37$ . This demonstrates the importance of using equivalent forms of functions to solve equations in mathematics.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by James Stewart
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Future Work

In future work, we can explore other functions and equations, and discuss how they can be rewritten in equivalent forms to facilitate their solution. We can also investigate the properties of exponents and how they can be used to solve equations. Additionally, we can explore the applications of mathematics in computer science and other fields.

Acknowledgments

The author would like to acknowledge the support of the mathematics department at XYZ University, and the guidance of Professor John Doe. The author would also like to thank the anonymous reviewers for their helpful comments and suggestions.

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Introduction

In our previous article, we explored the functions $f$ and $g$ , and discussed how they can be rewritten in equivalent forms to facilitate the solution of the equation $f(x)=g(x)$ . In this article, we will answer some of the most frequently asked questions about the functions $f$ and $g$ , and provide additional insights and explanations.

Q&A

Q: What is the difference between the functions $f$ and $g$ ?

A: The functions $f$ and $g$ are both exponential functions, but they have different bases and exponents. The function $f(x)$ has a base of 4 and an exponent of $5x-1$ , while the function $g(x)$ has a base of 8 and an exponent of $x/4$ .

Q: How do I rewrite the functions $f$ and $g$ in equivalent forms?

A: To rewrite the functions $f$ and $g$ in equivalent forms, you can use the properties of exponents. Specifically, you can rewrite $4^{(5x-1)}$ as $(2^2)^{(5x-1)}$ , and $8^{(x/4)}$ as $(2^3)^{(x/4)}$ . This will give you the equivalent forms $f(x)=2^{10x-2}$ and $g(x)=2^{(3x)/4}$ .

Q: How do I solve the equation $f(x)=g(x)$ ?

A: To solve the equation $f(x)=g(x)$ , you can set the two functions equal to each other and solve for $x$ . This will give you the equation $2^{10x-2}=2^{(3x)/4}$ . You can then use the properties of exponents to solve for $x$ .

Q: What is the solution to the equation $f(x)=g(x)$ ?

A: The solution to the equation $f(x)=g(x)$ is $x=8/37$ .

Q: Can I use the functions $f$ and $g$ to model real-world phenomena?

A: Yes, the functions $f$ and $g$ can be used to model real-world phenomena. For example, the function $f(x)$ can be used to model population growth, while the function $g(x)$ can be used to model the spread of a disease.

Q: Are there any limitations to the functions $f$ and $g$ ?

A: Yes, there are limitations to the functions $f$ and $g$ . For example, the function $f(x)$ is only defined for $x>1/5$ , while the function $g(x)$ is only defined for $x>0$ .

Additional Insights

The functions $f$ and $g$ are both examples of exponential functions, which are used to model growth and decay in a wide range of real-world phenomena.
The properties of exponents are essential for solving equations involving exponential functions.
The functions $f$ and $g$ can be used to model population growth, the spread of a disease, and other real-world phenomena.

Conclusion

In this article, we answered some of the most frequently asked questions about the functions $f$ and $g$ , and provided additional insights and explanations. We hope that this article has been helpful in clarifying the concepts and ideas presented in our previous article.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by James Stewart
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton