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

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Introduction

In this article, we will focus on solving the exponential equation $3^{x-5}=9$. This type of equation involves an exponential function with a variable in the exponent. To solve it, we need to isolate the variable and use properties of exponents. We will also explore the concept of exponential functions and their properties.

Understanding Exponential Functions

Exponential functions are a type of mathematical function that involves a base raised to a power. The general form of an exponential function is $f(x) = a^x$, where $a$ is the base and $x$ is the exponent. In the equation $3^{x-5}=9$, the base is $3$ and the exponent is $x-5$.

Properties of Exponents

To solve the equation $3^{x-5}=9$, we need to use properties of exponents. One of the most important properties of exponents is the power rule, which states that $(a^m)^n = a^{mn}$. We can also use the property that $a^m \cdot a^n = a^{m+n}$.

Solving the Equation

To solve the equation $3^{x-5}=9$, we can start by rewriting $9$ as a power of $3$. Since $9 = 3^2$, we can rewrite the equation as $3^{x-5}=3^2$. Now, we can use the property that if $a^m = a^n$, then $m = n$. Therefore, we can set the exponents equal to each other and solve for $x$.

Setting the Exponents Equal

Setting the exponents equal, we get $x-5 = 2$. Now, we can solve for $x$ by adding $5$ to both sides of the equation.

Solving for $x$

Solving for $x$, we get $x = 2 + 5$. Therefore, $x = 7$.

Conclusion

In this article, we solved the exponential equation $3^{x-5}=9$ by using properties of exponents. We started by rewriting $9$ as a power of $3$ and then used the property that if $a^m = a^n$, then $m = n$. We set the exponents equal and solved for $x$ by adding $5$ to both sides of the equation. The solution to the equation is $x = 7$.

Final Answer

The final answer to the equation $3^{x-5}=9$ is $x = 7$.

Discussion

The equation $3^{x-5}=9$ is a classic example of an exponential equation. Exponential equations involve an exponential function with a variable in the exponent. To solve them, we need to use properties of exponents and isolate the variable. In this article, we used the power rule and the property that if $a^m = a^n$, then $m = n$ to solve the equation. The solution to the equation is $x = 7$.

Related Topics

• Exponential functions
• Properties of exponents
• Solving exponential equations

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Properties of Exponents" by Mathway
• [3] "Solving Exponential Equations" by Khan Academy

Additional Resources

• [1] "Exponential Functions" by Wolfram MathWorld
• [2] "Properties of Exponents" by Purplemath
• [3] "Solving Exponential Equations" by MIT OpenCourseWare
  

Introduction

In our previous article, we solved the exponential equation $3^{x-5}=9$ by using properties of exponents. In this article, we will answer some frequently asked questions about solving exponential equations.

Q: What is an exponential equation?

A: An exponential equation is a type of equation that involves an exponential function with a variable in the exponent. The general form of an exponential equation is $a^x = b$, where $a$ is the base, $x$ is the exponent, and $b$ is the result.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable and use properties of exponents. You can start by rewriting the equation in a form that is easier to work with, such as rewriting $9$ as a power of $3$. Then, you can use the property that if $a^m = a^n$, then $m = n$ to set the exponents equal and solve for $x$.

Q: What are some common properties of exponents that I need to know?

A: Some common properties of exponents that you need to know include:

• The power rule: $(a^m)^n = a^{mn}$
• The product rule: $a^m \cdot a^n = a^{m+n}$
• The quotient rule: $\frac{a^m}{a^n} = a^{m-n}$
• The property that if $a^m = a^n$, then $m = n$

Q: How do I handle negative exponents?

A: To handle negative exponents, you can rewrite the equation in a form that is easier to work with. For example, if you have the equation $3^{-x} = 9$, you can rewrite it as $3^x = \frac{1}{9}$.

Q: Can I use logarithms to solve exponential equations?

A: Yes, you can use logarithms to solve exponential equations. In fact, logarithms are a powerful tool for solving exponential equations. You can use the property that $\log_a a^x = x$ to solve for $x$.

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

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

• Not rewriting the equation in a form that is easier to work with
• Not using the correct properties of exponents
• Not checking your work to make sure that the solution is correct

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

A: Yes, you can use a calculator to solve exponential equations. In fact, calculators are a great tool for checking your work and making sure that the solution is correct.

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

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

• Modeling population growth
• Modeling chemical reactions
• Modeling financial growth
• Modeling electrical circuits

Conclusion

In this article, we answered some frequently asked questions about solving exponential equations. We covered topics such as the definition of an exponential equation, how to solve an exponential equation, common properties of exponents, handling negative exponents, using logarithms, common mistakes to avoid, and real-world applications of exponential equations.

Final Answer

The final answer to the question "How do I solve an exponential equation?" is to use properties of exponents and isolate the variable.

Discussion

Solving exponential equations is an important topic in mathematics. Exponential equations have many real-world applications, and understanding how to solve them is crucial for many fields of study. In this article, we provided a comprehensive overview of how to solve exponential equations and answered some frequently asked questions.

Related Topics

• Exponential functions
• Properties of exponents
• Solving exponential equations
• Logarithms
• Real-world applications of exponential equations

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Properties of Exponents" by Mathway
• [3] "Solving Exponential Equations" by Khan Academy
• [4] "Logarithms" by Wolfram MathWorld
• [5] "Real-world Applications of Exponential Equations" by MIT OpenCourseWare

Additional Resources

• [1] "Exponential Functions" by Purplemath
• [2] "Properties of Exponents" by IXL
• [3] "Solving Exponential Equations" by Mathway
• [4] "Logarithms" by Khan Academy
• [5] "Real-world Applications of Exponential Equations" by Wolfram Alpha