Solve The Equation: $ 25^x = 125^{x+3} }$Choose The Correct Value Of { X $}$ A. { X = -10 $ $ B. { X = -9 $}$ C. { X = -3 $}$ D. { X = 3 $}$

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 the equation $25^x = 125^{x+3}$ and choose the correct value of $x$ from the given options.

Understanding Exponential Equations

Exponential equations involve variables raised to a power, and the goal is to solve for the variable. In this case, we have the equation $25^x = 125^{x+3}$. To solve this equation, we need to use the properties of exponents and logarithms.

Using the Properties of Exponents

The first step in solving the equation is to use the properties of exponents. We can rewrite the equation as:

$$25^x = (5^2)^x = 5^{2x}$$

and

$$125^{x+3} = (5^3)^{x+3} = 5^{3(x+3)}$$

Now, we can equate the two expressions:

$$5^{2x} = 5^{3(x+3)}$$

Using the One-to-One Property of Exponents

The one-to-one property of exponents states that if $a^x = a^y$, then $x = y$. We can use this property to solve the equation:

$$5^{2x} = 5^{3(x+3)}$$

$$2x = 3(x+3)$$

Solving for x

Now, we can solve for $x$:

$$2x = 3x + 9$$

$$-x = 9$$

$$x = -9$$

Conclusion

In this article, we solved the equation $25^x = 125^{x+3}$ and chose the correct value of $x$ from the given options. We used the properties of exponents and logarithms to simplify the equation and solve for $x$. The correct value of $x$ is $-9$.

Choosing the Correct Answer

Based on our solution, we can choose the correct answer from the given options:

• A. $x = -10$
• B. $x = -9$
• C. $x = -3$
• D. $x = 3$

The correct answer is:

• B. $x = -9$

Final Answer

The final answer is $\boxed{-9}$.

Additional Tips and Tricks

• When solving exponential equations, it's essential to use the properties of exponents and logarithms.
• The one-to-one property of exponents is a powerful tool for solving exponential equations.
• Always simplify the equation before solving for the variable.
• Check your answer by plugging it back into the original equation.

Common Mistakes to Avoid

• Not using the properties of exponents and logarithms.
• Not simplifying the equation before solving for the variable.
• Not checking the answer by plugging it back into the original equation.

Real-World Applications

Exponential equations have numerous real-world applications, including:

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

Conclusion

$$$$$$

Introduction

In our previous article, we solved the equation $25^x = 125^{x+3}$ and chose the correct value of $x$ from the given options. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving exponential equations.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable raised to a power. For example, $2^x = 8$ is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to use the properties of exponents and logarithms. You can start by simplifying the equation using the properties of exponents, and then use logarithms to solve for the variable.

Q: What is the one-to-one property of exponents?

A: The one-to-one property of exponents states that if $a^x = a^y$, then $x = y$. This property is a powerful tool for solving exponential equations.

Q: How do I use the one-to-one property of exponents?

A: To use the one-to-one property of exponents, you need to equate the two expressions in the equation and then solve for the variable. For example, if you have the equation $2^x = 2^y$, you can use the one-to-one property to conclude that $x = y$.

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

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

• Not using the properties of exponents and logarithms.
• Not simplifying the equation before solving for the variable.
• Not checking the answer by plugging it back into the original equation.

Q: How do I check my answer when solving an exponential equation?

A: To check your answer when solving an exponential equation, you need to plug it back into the original equation and verify that it is true. For example, if you have the equation $2^x = 8$ and you solve for $x$ to get $x = 3$, you can plug $x = 3$ back into the original equation to verify that it is true.

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

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

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

Q: How do I use logarithms to solve exponential equations?

A: To use logarithms to solve exponential equations, you need to take the logarithm of both sides of the equation and then use the properties of logarithms to solve for the variable. For example, if you have the equation $2^x = 8$, you can take the logarithm of both sides to get $x = \log_2(8)$.

Q: What are some common types of exponential equations?

A: Some common types of exponential equations include:

• Equations with the same base: $2^x = 2^y$
• Equations with different bases: $2^x = 3^y$
• Equations with exponents: $2^{x+3} = 3^y$

Q: How do I solve exponential equations with the same base?

A: To solve exponential equations with the same base, you need to use the one-to-one property of exponents. For example, if you have the equation $2^x = 2^y$, you can use the one-to-one property to conclude that $x = y$.

Q: How do I solve exponential equations with different bases?

A: To solve exponential equations with different bases, you need to use logarithms. For example, if you have the equation $2^x = 3^y$, you can take the logarithm of both sides to get $x = \log_2(3^y)$.

Conclusion

Solving exponential equations requires a deep understanding of the underlying principles. By using the properties of exponents and logarithms, we can simplify the equation and solve for the variable. In this article, we provided a Q&A guide to help you better understand the concepts and techniques involved in solving exponential equations.