If $343^x=49^{4-x}$, What Is The Value Of $x$?A. $\frac{5}{8}$ B. \$\frac{8}{5}$[/tex\] C. 2 D. 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 explore how to solve exponential equations, using the given equation $343^x=49{4-x}$ as a case study. We will break down the solution into manageable steps, making it easy to follow and understand.

Understanding Exponential Equations

Exponential equations involve variables in the exponent, and they can be solved using various techniques, including logarithms, algebraic manipulations, and properties of exponents. The general form of an exponential equation is $a^x=by$, where $a$ and $b$ are positive real numbers, and $x$ and $y$ are variables.

Properties of Exponents

Before we dive into solving the given equation, let's review some essential properties of exponents:

• Product of Powers: $a^m \cdot a^n = a^{m+n}$
• Power of a Power: $(a^m)n = a^{mn}$
• Quotient of Powers: $\frac{a^m}{an} = a^{m-n}$

Solving the Given Equation

Now that we have a solid understanding of exponential equations and properties of exponents, let's tackle the given equation $343^x=49{4-x}$.

Step 1: Rewrite the Equation

We can rewrite the equation using the fact that $343=7^3$ and $49=7^2$.

$$\begin{align*} (7^3)^x & = (7^2)^{4-x} \\ 7^{3x} & = 7^{8-2x} \end{align*}$$

Step 2: Equate the Exponents

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

$$\begin{align*} 3x & = 8-2x \\ 5x & = 8 \end{align*}$$

Step 3: Solve for x

Now that we have a linear equation, we can solve for $x$.

$$\begin{align*} x & = \frac{8}{5} \end{align*}$$

Conclusion

In this article, we have demonstrated how to solve exponential equations using algebraic manipulations and properties of exponents. We have used the given equation $343^x=49{4-x}$ as a case study, breaking down the solution into manageable steps. By following these steps, we have arrived at the solution $x = \frac{8}{5}$.

Discussion

The solution to the given equation is $x = \frac{8}{5}$. This value satisfies the original equation, and it can be verified by plugging it back into the equation.

Comparison of Options

Let's compare the solution with the given options:

• A. $\frac{5}{8}$: This is not the correct solution.
• B. $\frac{8}{5}$: This is the correct solution.
• C. 2: This is not the correct solution.
• D. 8: This is not the correct solution.

Final Answer

Introduction

In our previous article, we explored how to solve exponential equations using algebraic manipulations and properties of exponents. We used the given equation $343^x=49{4-x}$ as a case study, breaking down the solution into manageable steps. In this article, we will address some frequently asked questions related to solving exponential equations.

Q&A

Q: What is the first step in solving an exponential equation?

A: The first step in solving an exponential equation is to rewrite the equation using the properties of exponents. This involves simplifying the equation and making it easier to work with.

Q: How do I know when to use the product of powers property?

A: You should use the product of powers property when you have a product of two or more exponential expressions with the same base. For example, $a^m \cdot a^n = a^{m+n}$.

Q: Can I use the power of a power property to simplify an exponential expression?

A: Yes, you can use the power of a power property to simplify an exponential expression. For example, $(a^m)n = a^{mn}$.

Q: How do I solve an exponential equation with a variable in the exponent?

A: To solve an exponential equation with a variable in the exponent, you need to isolate the variable by using algebraic manipulations and properties of exponents. This may involve rewriting the equation, equating the exponents, and solving for the variable.

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

A: An exponential equation involves variables in the exponent, while a logarithmic equation involves variables as the base of the logarithm. For example, $a^x=b$ is an exponential equation, while $\log_a b = x$ is a logarithmic equation.

Q: Can I use a calculator to solve an exponential equation?

A: Yes, you can use a calculator to solve an exponential equation. However, it's always a good idea to check your work by plugging the solution 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:

• Not rewriting the equation using the properties of exponents
• Not equating the exponents when the bases are the same
• Not checking the solution by plugging it back into the original equation

Conclusion

In this article, we have addressed some frequently asked questions related to solving exponential equations. We have provided step-by-step solutions to common problems and highlighted some common mistakes to avoid. By following these tips and techniques, you can become more confident and proficient in solving exponential equations.

Additional Resources

For more information on solving exponential equations, we recommend the following resources:

• Khan Academy: Exponential Equations
• Mathway: Exponential Equations
• Wolfram Alpha: Exponential Equations

Final Tips

• Always rewrite the equation using the properties of exponents
• Equate the exponents when the bases are the same
• Check the solution by plugging it back into the original equation
• Use a calculator to check your work, but always verify the solution

By following these tips and techniques, you can become more confident and proficient in solving exponential equations. Happy solving!