Solve: $6^{2x-3} = 6^{-2x+1}$A. $x = -1$ B. $x = 0$ C. $x = 1$ D. $x = 4$

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulations and properties of exponents. In this article, we will focus on solving a specific exponential equation, $6^{2x-3} = 6^{-2x+1}$, and explore the different methods and techniques used to find the solution.

Understanding Exponential Equations

Exponential equations involve variables in the exponent, and they can be solved using various methods, including algebraic manipulations, logarithmic properties, and graphical methods. The equation $6^{2x-3} = 6^{-2x+1}$ is a classic example of an exponential equation, where the base is 6 and the exponents are expressions involving the variable x.

Step 1: Simplify the Equation

To solve the equation $6^{2x-3} = 6^{-2x+1}$, we can start by simplifying the equation using the properties of exponents. We can rewrite the equation as:

$$6^{2x-3} = 6^{-2x+1}$$

$$\Rightarrow 6^{2x-3} \cdot 6^{2x-1} = 1$$

Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$, we can simplify the left-hand side of the equation:

$$6^{(2x-3)+(2x-1)} = 1$$

$$\Rightarrow 6^{4x-4} = 1$$

Step 2: Use Logarithmic Properties

Now that we have simplified the equation, we can use logarithmic properties to solve for x. We can take the logarithm of both sides of the equation using any base, but let's use the natural logarithm (ln) for simplicity:

$$\ln(6^{4x-4}) = \ln(1)$$

Using the property of logarithms that states $\ln(a^b) = b \cdot \ln(a)$, we can simplify the left-hand side of the equation:

$$(4x-4) \cdot \ln(6) = 0$$

Step 3: Solve for x

Now that we have simplified the equation, we can solve for x. We can start by isolating the term involving x:

$$(4x-4) \cdot \ln(6) = 0$$

$$\Rightarrow 4x-4 = 0$$

Adding 4 to both sides of the equation, we get:

$$4x = 4$$

Dividing both sides of the equation by 4, we get:

$$x = 1$$

Conclusion

In this article, we have solved the exponential equation $6^{2x-3} = 6^{-2x+1}$ using algebraic manipulations and logarithmic properties. We have shown that the solution to the equation is x = 1. This problem is a great example of how to use exponential equations and logarithmic properties to solve for variables.

Answer

The correct answer is C. $x = 1$.

Additional Tips and Tricks

• When solving exponential equations, it's essential to simplify the equation using the properties of exponents.
• Logarithmic properties can be used to solve for variables in exponential equations.
• Make sure to check your work and verify the solution by plugging it back into the original equation.

Real-World Applications

Exponential equations have numerous real-world applications, including:

• Modeling population growth and decay
• Analyzing financial data and predicting stock prices
• Understanding chemical reactions and kinetics
• Solving problems in physics and engineering

By mastering the techniques and methods used to solve exponential equations, you can apply them to a wide range of real-world problems and make informed decisions.

Final Thoughts

Introduction

In our previous article, we explored the concept of exponential equations and solved a specific equation using algebraic manipulations and logarithmic properties. In this article, we will provide a Q&A guide to help you better understand the topic and address any questions or concerns you may have.

Q: What is an exponential equation?

A: An exponential equation is a mathematical equation that involves a variable in the exponent. It is a type of equation that can be solved using various methods, including algebraic manipulations, logarithmic properties, and graphical methods.

Q: How do I simplify an exponential equation?

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

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

A: A logarithmic equation is an equation that involves a variable in the logarithm, while an exponential equation is an equation that involves a variable in the exponent. Logarithmic equations can be solved using logarithmic properties, while exponential equations can be solved using algebraic manipulations and logarithmic properties.

Q: How do I use logarithmic properties to solve an exponential equation?

A: To use logarithmic properties to solve an exponential equation, you can take the logarithm of both sides of the equation using any base. For example, if you have the equation $a^x = b$, you can take the logarithm of both sides to get $x \cdot \log(a) = \log(b)$.

Q: What is the significance of the base in an exponential equation?

A: The base in an exponential equation is the number that is raised to the power of the variable. The base can be any positive number, and it determines the rate at which the variable grows or decays.

Q: How do I verify the solution to an exponential equation?

A: To verify the solution to an exponential equation, you can plug it back into the original equation and check if it is true. For example, if you have the equation $a^x = b$ and you find that $x = c$, you can plug $c$ back into the equation to get $a^c = b$.

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

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

• Modeling population growth and decay
• Analyzing financial data and predicting stock prices
• Understanding chemical reactions and kinetics
• Solving problems in physics and engineering

Q: How can I practice solving exponential equations?

A: You can practice solving exponential equations by working through examples and exercises in a textbook or online resource. You can also try solving real-world problems that involve exponential equations.

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 using the properties of exponents
• Not using logarithmic properties to solve the equation
• Not verifying the solution by plugging it back into the original equation

Conclusion

In this Q&A guide, we have addressed some common questions and concerns about solving exponential equations. We hope that this guide has been helpful in clarifying the concept and providing you with a better understanding of the topic. Remember to practice solving exponential equations and to verify your solutions to ensure accuracy.

Additional Resources

• Online resources: Khan Academy, Mathway, Wolfram Alpha
• Textbooks: Algebra and Trigonometry by Michael Sullivan, Calculus by Michael Spivak
• Practice problems: Exponential Equations Worksheet, Exponential Growth and Decay Problems

Final Thoughts

Solving exponential equations requires a deep understanding of algebraic manipulations and logarithmic properties. By following the steps outlined in this guide, you can solve complex exponential equations and apply them to real-world problems. Remember to simplify the equation, use logarithmic properties, and verify the solution to ensure accuracy. With practice and patience, you can become proficient in solving exponential equations and tackle even the most challenging problems.