Solve The Equation:$\[ 81^{2x+1} = 27^{x-2} \\]

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of exponential functions and their properties. In this article, we will focus on solving the equation $81^{2x+1} = 27^{x-2}$, which involves manipulating exponential expressions and using algebraic techniques to isolate the variable.

Understanding Exponential Functions

Before we dive into solving the equation, let's take a moment to understand the properties of exponential functions. An exponential function is a function of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable. The base $a$ determines the rate at which the function grows or decays.

In the given equation, we have two exponential expressions: $81^{2x+1}$ and $27^{x-2}$. To solve this equation, we need to manipulate these expressions using algebraic techniques and properties of exponents.

Simplifying the Equation

The first step in solving the equation is to simplify the expressions on both sides. We can start by expressing both bases as powers of 3, since $81 = 3^4$ and $27 = 3^3$.

$$81^{2x+1} = (3^4)^{2x+1} = 3^{8x+4}$$

$$27^{x-2} = (3^3)^{x-2} = 3^{3x-6}$$

Now we have the equation:

$$3^{8x+4} = 3^{3x-6}$$

Using Properties of Exponents

Since the bases are the same, we can equate the exponents:

$$8x+4 = 3x-6$$

Solving for x

Now we have a linear equation in one variable. We can solve for $x$ by isolating the variable on one side of the equation.

$$8x - 3x = -6 - 4$$

$$5x = -10$$

$$x = -2$$

Verifying the Solution

To verify our solution, we can plug $x = -2$ back into the original equation and check if it holds true.

$$81^{2(-2)+1} = 27^{(-2)-2}$$

$$81^{-3} = 27^{-4}$$

$$\frac{1}{81^3} = \frac{1}{27^4}$$

Since the two expressions are equal, we have verified that $x = -2$ is indeed the solution to the equation.

Conclusion

Solving exponential equations requires a deep understanding of exponential functions and their properties. By simplifying the expressions, using properties of exponents, and solving for the variable, we can find the solution to the equation $81^{2x+1} = 27^{x-2}$. In this article, we have walked through the step-by-step process of solving this equation, and we have verified that the solution is $x = -2$.

Tips and Tricks

• When solving exponential equations, it's essential to simplify the expressions using properties of exponents.
• If the bases are the same, you can equate the exponents and solve for the variable.
• Always verify your solution by plugging it back into the original equation.

Common Mistakes

• Failing to simplify the expressions using properties of exponents.
• Not equating the exponents when the bases are the same.
• Not verifying the solution by plugging it back into the original equation.

Real-World Applications

Exponential equations have numerous real-world applications in fields such as finance, economics, and science. For example, exponential growth and decay are used to model population growth, chemical reactions, and financial investments.

Practice Problems

Try solving the following exponential equations:

• $2^{3x-2} = 4^{x+1}$
• $5^{2x+1} = 25^{x-2}$
• $3^{x-2} = 9^{2x+1}$

References

• [1] "Exponential Functions" by Khan Academy
• [2] "Solving Exponential Equations" by Mathway
• [3] "Exponential Growth and Decay" by Wolfram MathWorld
  Solving Exponential Equations: A Q&A Guide =====================================================

Introduction

In our previous article, we walked through the step-by-step process of solving the equation $81^{2x+1} = 27^{x-2}$. In this article, we will answer some of the most frequently asked questions about solving exponential equations.

Q: What is an exponential equation?

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

Q: How do I simplify exponential expressions?

A: To simplify exponential expressions, you can use the properties of exponents, such as the product rule, the quotient rule, and the power rule. For example, if you have the expression $a^{m+n}$, you can simplify it by using the product rule: $a^{m+n} = a^m \cdot a^n$.

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

A: An exponential equation involves an exponential expression, while a logarithmic equation involves a logarithmic expression. For example, the equation $2^x = 8$ is an exponential equation, while the equation $\log_2 8 = x$ is a logarithmic equation.

Q: How do I solve an exponential equation?

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

1. Simplify the exponential expressions using the properties of exponents.
2. Equate the exponents if the bases are the same.
3. Solve for the variable.
4. Verify the solution by plugging it back into the original equation.

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

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

• Failing to simplify the exponential expressions using the properties of exponents.
• Not equating the exponents if the bases are the same.
• Not verifying the solution by plugging it back into the original equation.

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

A: Exponential equations have numerous real-world applications in fields such as finance, economics, and science. For example, exponential growth and decay are used to model population growth, chemical reactions, and financial investments.

Q: What are some examples of exponential equations in real-world situations?

A: Some examples of exponential equations in real-world situations include:

• Population growth: The population of a city grows exponentially, with a growth rate of 2% per year.
• Chemical reactions: The concentration of a chemical in a solution decreases exponentially over time.
• Financial investments: The value of a stock increases exponentially over time, with a growth rate of 10% per year.

Q: How do I practice solving exponential equations?

A: To practice solving exponential equations, you can try the following:

• Solve exponential equations on your own using the steps outlined above.
• Use online resources, such as Khan Academy or Mathway, to practice solving exponential equations.
• Work with a partner or tutor to practice solving exponential equations.

Q: What are some resources for learning more about exponential equations?

A: Some resources for learning more about exponential equations include:

• Khan Academy: Exponential Functions
• Mathway: Solving Exponential Equations
• Wolfram MathWorld: Exponential Growth and Decay

Conclusion

Solving exponential equations requires a deep understanding of exponential functions and their properties. By following the steps outlined above and practicing solving exponential equations, you can become proficient in solving these types of equations. Remember to simplify the exponential expressions, equate the exponents if the bases are the same, and verify the solution by plugging it back into the original equation.