For What Value Of $x$ Does $3^{4x} = 27^{x-3}$?A. − 9 -9 − 9 B. − 3 -3 − 3 C. 3 3 3 D. 9 9 9

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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 solving a specific type of exponential equation, where the bases are the same but the exponents are different. We will use the given equation $3^{4x} = 27^{x-3}$ as a case study to demonstrate the step-by-step process of solving exponential equations.

Understanding the Equation

The given equation is $3^{4x} = 27^{x-3}$. To begin solving this equation, we need to understand the properties of exponents and the relationship between the bases. The base of an exponential expression is the number that is raised to a power, while the exponent is the power to which the base is raised.

In this equation, the base is 3, and the exponents are 4x4x and x−3x-3. We can rewrite 27 as 333^3, so the equation becomes $3^{4x} = (33){x-3}$. Using the property of exponents that (am)n=amn(a^m)^n = a^{mn}, we can simplify the right-hand side of the equation to $3^{3(x-3)}$.

Simplifying the Equation

Now that we have simplified the right-hand side of the equation, we can rewrite the equation as $3^{4x} = 3^{3(x-3)}$. Since the bases are the same, we can equate the exponents, which gives us the equation $4x = 3(x-3)$.

Solving the Linear Equation

The equation $4x = 3(x-3)$ is a linear equation in one variable. To solve for xx, we can expand the right-hand side of the equation and then isolate the variable. Expanding the right-hand side gives us $4x = 3x - 9$.

Isolating the Variable

To isolate the variable xx, we need to get all the terms with xx on one side of the equation. We can do this by subtracting 3x3x from both sides of the equation, which gives us $4x - 3x = -9$. Simplifying the left-hand side gives us $x = -9$.

Conclusion

In this article, we have demonstrated the step-by-step process of solving an exponential equation. We started with the given equation $3^{4x} = 27^{x-3}$ and simplified it to $3^{4x} = 3^{3(x-3)}$. We then equated the exponents and solved the resulting linear equation to find the value of xx. The final answer is x=−9x = -9.

Final Answer

The final answer is x=−9x = -9. This is the value of xx that satisfies the given equation.

Discussion

The given equation is a classic example of an exponential equation, where the bases are the same but the exponents are different. Solving this equation requires a deep understanding of the properties of exponents and the relationship between the bases. The step-by-step process demonstrated in this article provides a clear and concise guide to solving exponential equations.

Related Topics

  • Exponential functions
  • Properties of exponents
  • Linear equations
  • Algebraic manipulations

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for the Nonmathematician" by Morris Kline

Keywords

  • Exponential equations
  • Properties of exponents
  • Linear equations
  • Algebraic manipulations
  • Exponential functions
  • Mathematics
  • Algebra
  • Calculus

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Introduction

Exponential equations can be a challenging topic for many students, but with practice and patience, anyone can master them. In this article, we will answer some of the most frequently asked questions about exponential equations, providing a clear and concise guide to help you understand this important concept.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power. For example, the equation $3^{4x} = 27^{x-3}$ is an exponential equation because it involves the exponential expression $3^{4x}$.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to follow these steps:

  1. Simplify the equation by rewriting the bases as the same number.
  2. Equate the exponents.
  3. Solve the resulting linear equation.

Q: What is the difference between an exponential equation and a linear equation?

A: An exponential equation involves an exponential expression, while a linear equation involves a linear expression. For example, the equation $2x + 3 = 5$ is a linear equation, while the equation $2^{3x} = 8^{x-2}$ is an exponential equation.

Q: Can I use the same methods to solve exponential equations as I would to solve linear equations?

A: No, you cannot use the same methods to solve exponential equations as you would to solve linear equations. Exponential equations require a different set of techniques and strategies, such as simplifying the equation and equating the exponents.

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

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

  • Not simplifying the equation before equating the exponents.
  • Not equating the exponents correctly.
  • Not solving the resulting linear equation correctly.

Q: Can I use a calculator to solve exponential equations?

A: Yes, you can use a calculator to solve exponential equations, but it's not always the best approach. Calculators can be useful for checking your work or for solving equations that are too complex to solve by hand. However, it's always a good idea to try to solve the equation by hand first, as this will help you understand the underlying concepts and techniques.

Q: What are some real-world applications of exponential equations?

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

  • Modeling population growth and decay.
  • Calculating compound interest.
  • Analyzing the spread of diseases.
  • Predicting the behavior of complex systems.

Q: Can I use exponential equations to solve problems in other areas of mathematics?

A: Yes, exponential equations can be used to solve problems in other areas of mathematics, including:

  • Algebra.
  • Calculus.
  • Statistics.
  • Probability.

Q: What are some common types of exponential equations?

A: Some common types of exponential equations include:

  • Equations with the same base.
  • Equations with different bases.
  • Equations with negative exponents.
  • Equations with fractional exponents.

Q: Can I use exponential equations to solve problems in science and engineering?

A: Yes, exponential equations can be used to solve problems in science and engineering, including:

  • Modeling the behavior of physical systems.
  • Analyzing the spread of diseases.
  • Predicting the behavior of complex systems.
  • Calculating the growth of populations.

Q: What are some common mistakes to avoid when using exponential equations in science and engineering?

A: Some common mistakes to avoid when using exponential equations in science and engineering include:

  • Not understanding the underlying concepts and techniques.
  • Not using the correct mathematical models.
  • Not considering the limitations of the mathematical models.
  • Not checking the results for accuracy.

Q: Can I use exponential equations to solve problems in finance and economics?

A: Yes, exponential equations can be used to solve problems in finance and economics, including:

  • Calculating compound interest.
  • Modeling the growth of investments.
  • Analyzing the behavior of financial markets.
  • Predicting the behavior of economic systems.

Q: What are some common types of exponential equations used in finance and economics?

A: Some common types of exponential equations used in finance and economics include:

  • Equations with the same base.
  • Equations with different bases.
  • Equations with negative exponents.
  • Equations with fractional exponents.

Q: Can I use exponential equations to solve problems in other areas of science and engineering?

A: Yes, exponential equations can be used to solve problems in other areas of science and engineering, including:

  • Physics.
  • Chemistry.
  • Biology.
  • Computer science.

Q: What are some common mistakes to avoid when using exponential equations in other areas of science and engineering?

A: Some common mistakes to avoid when using exponential equations in other areas of science and engineering include:

  • Not understanding the underlying concepts and techniques.
  • Not using the correct mathematical models.
  • Not considering the limitations of the mathematical models.
  • Not checking the results for accuracy.

Conclusion

Exponential equations are a fundamental concept in mathematics, and they have many real-world applications. By understanding the underlying concepts and techniques, you can use exponential equations to solve problems in a variety of fields, including science, engineering, finance, and economics. Remember to avoid common mistakes, such as not simplifying the equation before equating the exponents, and not solving the resulting linear equation correctly. With practice and patience, you can master exponential equations and use them to solve complex problems.