Solve For \[$ X \$\]:$\[ 6^{5x+4} = 6^{-10+7x} \\]

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of exponents. In this article, we will focus on solving a specific type of exponential equation, where the base is the same on both sides of the equation. We will use the given equation as an example and walk through the step-by-step process of solving it.

The Given Equation

The given equation is:

$$6^{5x+4} = 6^{-10+7x}$$

Our goal is to solve for the variable $x$.

Step 1: Set the Exponents Equal

Since the bases are the same on both sides of the equation, we can set the exponents equal to each other:

$$5x+4 = -10+7x$$

Step 2: Simplify the Equation

Now, let's simplify the equation by combining like terms:

$$5x+4 = -10+7x$$

Subtracting $5x$ from both sides gives us:

$$4 = -10+2x$$

Adding $10$ to both sides gives us:

$$14 = 2x$$

Step 3: Solve for $x$

Now, let's solve for $x$ by dividing both sides of the equation by $2$:

$$x = \frac{14}{2}$$

Simplifying the fraction gives us:

$$x = 7$$

Conclusion

In this article, we solved a specific type of exponential equation, where the base is the same on both sides of the equation. We used the given equation as an example and walked through the step-by-step process of solving it. By setting the exponents equal, simplifying the equation, and solving for $x$, we were able to find the value of the variable.

Tips and Tricks

• When solving exponential equations, make sure to set the exponents equal to each other, since the bases are the same on both sides of the equation.
• Simplify the equation by combining like terms and isolating the variable.
• Use algebraic properties, such as addition and subtraction, to simplify the equation.

Real-World Applications

Exponential equations have many real-world applications, such as:

• Modeling population growth and decay
• Calculating compound interest
• Analyzing chemical reactions

Common Mistakes

• Failing to set the exponents equal to each other
• Not simplifying the equation properly
• Not isolating the variable

Practice Problems

Try solving the following exponential equations:

• $2^{3x-2} = 2^{5x+1}$
• $3^{2x+1} = 3^{x-4}$
• $4^{x-2} = 4^{2x+3}$

Solutions

• $x = \frac{3}{2}$
• $x = -\frac{5}{3}$
• $x = -\frac{11}{5}$

Conclusion

Introduction

In our previous article, we discussed the step-by-step process of solving exponential equations. In this article, we will address some common questions and concerns that students may have when solving exponential equations.

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

A: The first step in solving an exponential equation is to set the exponents equal to each other, since the bases are the same on both sides of the equation.

Q: How do I simplify the equation after setting the exponents equal?

A: After setting the exponents equal, you can simplify the equation by combining like terms and isolating the variable. This may involve adding, subtracting, multiplying, or dividing both sides of the equation.

Q: What if the equation has a negative exponent?

A: If the equation has a negative exponent, you can rewrite it as a positive exponent by moving the base to the other side of the equation. For example, if the equation is $2^{-3x} = 4$, you can rewrite it as $2^{3x} = 4$.

Q: Can I use logarithms to solve exponential equations?

A: Yes, you can use logarithms to solve exponential equations. In fact, logarithms can be a powerful tool for solving exponential equations, especially when the exponents are large or complex.

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

A: An exponential equation is an equation that involves an exponential expression, such as $2^{3x} = 8$. A logarithmic equation, on the other hand, is an equation that involves a logarithmic expression, such as $\log_{2} 8 = 3$.

Q: How do I know which base to use when solving an exponential equation?

A: When solving an exponential equation, you can use any base that is convenient for the problem. However, it's often easiest to use a base that is a power of 10, such as 2 or 10.

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

A: Yes, you can use a calculator to solve exponential equations. In fact, calculators can be a powerful tool for solving exponential equations, especially when the exponents are large or complex.

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

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

• Failing to set the exponents equal to each other
• Not simplifying the equation properly
• Not isolating the variable
• Using the wrong base or exponent

Q: How can I practice solving exponential equations?

A: You can practice solving exponential equations by working through example problems, such as those found in this article. You can also try solving exponential equations on your own, using a calculator or other tools as needed.

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

A: Exponential equations have many real-world applications, such as:

• Modeling population growth and decay
• Calculating compound interest
• Analyzing chemical reactions
• Predicting the spread of diseases

Conclusion

In conclusion, solving exponential equations requires a deep understanding of the properties of exponents and algebraic properties. By following the step-by-step process outlined in this article, you can solve exponential equations with ease. Remember to set the exponents equal, simplify the equation, and solve for the variable. With practice, you will become proficient in solving exponential equations and be able to apply them to real-world problems.

Additional Resources

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

Practice Problems

Try solving the following exponential equations:

• $2^{3x-2} = 2^{5x+1}$
• $3^{2x+1} = 3^{x-4}$
• $4^{x-2} = 4^{2x+3}$

Solutions

• $x = \frac{3}{2}$
• $x = -\frac{5}{3}$
• $x = -\frac{11}{5}$