Solve For { X $} : : : { 4^{x-5} = 8^{2x-2} \} Answer Attempt 1 Out Of 3:${ X = \square }$

Introduction

Exponential equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving the equation $4^{x-5} = 8^{2x-2}$ to find the value of $x$. We will break down the solution into manageable steps, making it easier to understand and follow along.

Step 1: Simplify the Equation

The first step in solving the equation is to simplify it by expressing both sides with the same base. In this case, we can rewrite $8$ as $4^2$. This gives us:

$$4^{x-5} = (4^2)^{2x-2}$$

Using the property of exponents that $(a^b)^c = a^{bc}$, we can simplify the right-hand side of the equation:

$$4^{x-5} = 4^{2(2x-2)}$$

This simplifies to:

$$4^{x-5} = 4^{4x-8}$$

Step 2: Set the Exponents Equal

Since the bases are the same, we can set the exponents equal to each other:

$$x-5 = 4x-8$$

Step 3: Solve for x

Now, we can solve for $x$ by isolating it on one side of the equation. First, let's add $5$ to both sides:

$$x = 4x-3$$

Next, let's subtract $4x$ from both sides:

$$-3x = -3$$

Finally, let's divide both sides by $-3$:

$$x = 1$$

Conclusion

In this article, we solved the exponential equation $4^{x-5} = 8^{2x-2}$ to find the value of $x$. By simplifying the equation, setting the exponents equal, and solving for $x$, we arrived at the solution $x = 1$. This demonstrates the importance of breaking down complex equations into manageable steps and using the properties of exponents to simplify them.

Common Mistakes to Avoid

When solving exponential equations, it's essential to avoid common mistakes such as:

• Not simplifying the equation before solving
• Not setting the exponents equal when the bases are the same
• Not isolating the variable on one side of the equation

By being aware of these common mistakes, you can avoid them and arrive at the correct solution.

Real-World Applications

Exponential equations have numerous real-world applications, including:

• Modeling population growth and decline
• Calculating compound interest
• Analyzing the spread of diseases
• Understanding the behavior of chemical reactions

By understanding how to solve exponential equations, you can apply this knowledge to real-world problems and make informed decisions.

Practice Problems

To reinforce your understanding of solving exponential equations, try the following practice problems:

• Solve the equation $2^{x-3} = 4^{2x-1}$
• Solve the equation $3^{x-2} = 9^{2x-3}$
• Solve the equation $5^{x-1} = 25^{2x-2}$

By practicing these problems, you can develop your skills and become more confident in solving exponential equations.

Conclusion

Introduction

In our previous article, we explored the step-by-step process of solving exponential equations. However, we understand that sometimes, it's not enough to just follow a set of instructions. You may have questions, doubts, or uncertainties that need to be addressed. That's why we've put together this Q&A guide to help you better understand the concept of solving exponential equations.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a mathematical expression that represents a quantity that grows or decays at a constant rate. Exponential equations can be written in the form $a^x = b$, where $a$ is the base, $x$ is the exponent, and $b$ is the result.

Q: How do I know if an equation is exponential?

A: To determine if an equation is exponential, look for the presence of an exponent, which is typically represented by a small number raised to a power (e.g., $2^3$). If you see an exponent, it's likely an exponential equation.

Q: What are the properties of exponents?

A: The properties of exponents are essential to solving exponential equations. Some of the key properties include:

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

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, follow these steps:

1. Identify the base: Determine the base of the exponential expression.
2. Simplify the exponent: Simplify the exponent by combining like terms or using the properties of exponents.
3. Rewrite the equation: Rewrite the equation with the simplified exponent.

Q: What if the bases are different?

A: If the bases are different, you can't simply set the exponents equal. Instead, you need to use the properties of exponents to rewrite the equation with the same base. For example, if you have the equation $2^x = 3^y$, you can rewrite it as $(2^x) \cdot (3^y) = 1$, which can be simplified to $2^x \cdot 3^y = 1$.

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, follow these steps:

1. Isolate the variable: Isolate the variable by moving all other terms to one side of the equation.
2. Take the logarithm: Take the logarithm of both sides of the equation to eliminate the exponent.
3. Solve for the variable: Solve for the variable by isolating it on one side of the 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 simplifying the equation: Failing to simplify the equation before solving can lead to incorrect solutions.
• Not setting the exponents equal: Failing to set the exponents equal when the bases are the same can lead to incorrect solutions.
• Not using the properties of exponents: Failing to use the properties of exponents can lead to incorrect solutions.

Conclusion

Solving exponential equations requires a solid understanding of the properties of exponents and a step-by-step approach. By avoiding common mistakes and using the properties of exponents, you can become proficient in solving exponential equations. If you have any further questions or doubts, feel free to ask!