Solve 3 X − 5 = 9 3^{x-5}=9 3 X − 5 = 9 .A. X = − 3 X=-3 X = − 3 B. X = 2 X=2 X = 2 C. X = 7 X=7 X = 7 D. X = 8 X=8 X = 8

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 the equation $3^{x-5}=9$, which is a classic example of an exponential equation. We will break down the solution into manageable steps, making it easy for readers to follow along and understand the process.

Understanding Exponential Equations

Exponential equations involve variables in the exponent, and they can be solved using various techniques, including logarithmic properties and algebraic manipulations. The general form of an exponential equation is $a^x=b$, where $a$ and $b$ are constants, and $x$ is the variable. In our case, the equation is $3^{x-5}=9$, where $a=3$, $b=9$, and $x$ is the variable.

Step 1: Simplify the Equation

To solve the equation $3^{x-5}=9$, we need to simplify it by getting rid of the exponent. We can do this by using the property of exponents that states $a^{b+c}=a^ba^c$. In this case, we can rewrite the equation as $3^{x-5}=3^2$, since $9=3^2$.

import math
# Given equation
equation = "3^(x-5) = 9"
# Simplify the equation
simplified_equation = "3^(x-5) = 3^2"
print(simplified_equation)

Step 2: Equate the Exponents

Now that we have simplified the equation, we can equate the exponents. Since the bases are the same (both are 3), we can set the exponents equal to each other: $x-5=2$.

# Equate the exponents
exponents_equal = "x - 5 = 2"
print(exponents_equal)

Step 3: Solve for x

To solve for $x$, we need to isolate the variable. We can do this by adding 5 to both sides of the equation: $x=2+5$.

# Solve for x
x_value = "x = 2 + 5"
print(x_value)

Step 4: Evaluate the Expression

Now that we have solved for $x$, we can evaluate the expression: $x=7$.

# Evaluate the expression
x_value = 7
print(x_value)

Conclusion

In this article, we have solved the exponential equation $3^{x-5}=9$ using a step-by-step approach. We simplified the equation, equated the exponents, solved for $x$, and evaluated the expression. The final answer is $x=7$. This problem is a great example of how to solve exponential equations, and it demonstrates the importance of understanding algebraic manipulations and properties of exponents.

Answer

The correct answer is:

• A. $x=-3$: Incorrect
• B. $x=2$: Incorrect
• C. $x=7$: Correct
• D. $x=8$: Incorrect

Discussion

This problem is a great example of how to solve exponential equations. The key concept is to simplify the equation, equate the exponents, and solve for the variable. The properties of exponents, such as $a^{b+c}=a^ba^c$, are essential in solving these types of equations. If you have any questions or need further clarification, please feel free to ask in the comments section below.

Related Problems

If you want to practice solving exponential equations, here are some related problems:

• $2^{x+3}=16$
• $5^{x-2}=125$
• $3^{x+1}=27$

Introduction

In our previous article, we solved the exponential equation $3^{x-5}=9$ using a step-by-step approach. We simplified the equation, equated the exponents, solved for $x$, and evaluated the expression. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving exponential equations.

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 techniques, including logarithmic properties and algebraic manipulations.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you need to get rid of the exponent. You can do this by using the property of exponents that states $a^{b+c}=a^ba^c$. For example, if you have the equation $3^{x-5}=9$, you can rewrite it as $3^{x-5}=3^2$, since $9=3^2$.

Q: How do I equate the exponents?

A: To equate the exponents, you need to set the exponents equal to each other. Since the bases are the same (both are 3), you can set the exponents equal to each other: $x-5=2$.

Q: How do I solve for x?

A: To solve for $x$, you need to isolate the variable. You can do this by adding 5 to both sides of the equation: $x=2+5$.

Q: What is the final answer?

A: The final answer is $x=7$.

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 setting the exponents equal to each other
• Not isolating the variable
• Not evaluating the expression correctly

Q: How do I practice solving exponential equations?

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

• $2^{x+3}=16$
• $5^{x-2}=125$
• $3^{x+1}=27$

Try solving these problems using the same step-by-step approach we used in this article. Good luck!

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

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

• Modeling population growth
• Calculating compound interest
• Analyzing chemical reactions
• Predicting stock prices

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and techniques involved in solving exponential equations. We covered topics such as simplifying exponential equations, equating the exponents, solving for $x$, and evaluating the expression. We also provided some common mistakes to avoid and some real-world applications of exponential equations. If you have any questions or need further clarification, please feel free to ask in the comments section below.

Related Articles

If you want to learn more about exponential equations, here are some related articles:

• Solving Exponential Equations: A Step-by-Step Guide
• Exponential Equations: A Review of the Basics
• Solving Exponential Equations with Logarithms

Try reading these articles to learn more about exponential equations and how to solve them. Good luck!