Solve For \[$x\$\] In The Equation:$\[ 625^{2x+4} = 3125^{2x-4} \\]

Feb 27, 2025 by ADMIN 68 views

**Solving Exponential Equations: A Step-by-Step Guide**

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will focus on solving the equation $625^{2x+4} = 3125^{2x-4}$ for the variable $x$ . We will break down the solution into manageable steps, using a combination of algebraic manipulations and properties of exponents.

Understanding Exponents

Before we dive into the solution, let's take a moment to review the basics of exponents. An exponent is a small number that is placed above and to the right of a base number, indicating how many times the base number should be multiplied by itself. For example, in the expression $a^b$ , the base is $a$ and the exponent is $b$ . When we raise a number to a power, we are essentially multiplying the base by itself as many times as the exponent indicates.

Rewriting the Equation

To solve the equation $625^{2x+4} = 3125^{2x-4}$ , we need to rewrite it in a more manageable form. We can start by expressing both sides of the equation in terms of a common base. Since $625 = 5^4$ and $3125 = 5^5$ , we can rewrite the equation as:

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

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

5^{8x+16} = 5^{10x-20}

Equating Exponents

Now that we have rewritten the equation in a more manageable form, we can equate the exponents on both sides. Since the bases are the same, the exponents must be equal:

8x+16 = 10x-20

Solving for x

To solve for $x$ , we can isolate the variable on one side of the equation. Let's start by subtracting $8x$ from both sides:

16 = 2x-20

Next, we can add $20$ to both sides:

36 = 2x

Finally, we can divide both sides by $2$ to solve for $x$ :

x = 18

Conclusion

In this article, we have solved the equation $625^{2x+4} = 3125^{2x-4}$ for the variable $x$ . We started by rewriting the equation in a more manageable form, using the properties of exponents to simplify the expression. We then equated the exponents on both sides and solved for $x$ using basic algebraic manipulations. The final solution is $x = 18$ .

Tips and Tricks

When solving exponential equations, it's essential to remember the following tips and tricks:

Use the properties of exponents: Exponents have several useful properties that can help simplify complex expressions. For example, $(a^b)^c = a^{bc}$ and $a^{b+c} = a^b \cdot a^c$ .
Equating exponents: When the bases are the same, the exponents must be equal. This is a fundamental property of exponential equations.
Isolate the variable: To solve for the variable, isolate it on one side of the equation using basic algebraic manipulations.

By following these tips and tricks, you can become proficient in solving exponential equations and tackle even the most challenging problems with confidence.

Common Mistakes to Avoid

When solving exponential equations, it's easy to make mistakes. Here are some common pitfalls to avoid:

Forgetting to equate exponents: When the bases are the same, the exponents must be equal. Don't forget to equate the exponents on both sides of the equation.
Not using the properties of exponents: Exponents have several useful properties that can help simplify complex expressions. Don't be afraid to use them to your advantage.
Not isolating the variable: To solve for the variable, isolate it on one side of the equation using basic algebraic manipulations.

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

Real-World Applications

Exponential equations have numerous real-world applications in fields such as finance, economics, and computer science. Here are a few examples:

Compound interest: Exponential equations are used to calculate compound interest in finance. For example, if you deposit $1000 into a savings account with a 5% annual interest rate, the balance after one year will be $1050, after two years will be $1102.50, and so on.
Population growth: Exponential equations are used to model population growth in biology and ecology. For example, if a population grows at a rate of 2% per year, the population after one year will be 102% of the original population, after two years will be 104.04% of the original population, and so on.
Computer science: Exponential equations are used in computer science to model the growth of algorithms and data structures. For example, if an algorithm has a time complexity of O(n^2), the running time will grow exponentially with the size of the input.

By understanding exponential equations and how to solve them, you can gain a deeper appreciation for the underlying principles of mathematics and its applications in the real world.

Conclusion

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power. For example, the equation $2^x = 8$ is an exponential equation because it involves the exponential expression $2^x$ .

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable (in this case, $x$ ) by using algebraic manipulations and properties of exponents. Here are the general steps:

Rewrite the equation: Rewrite the equation in a more manageable form by expressing both sides of the equation in terms of a common base.
Equating exponents: Equate the exponents on both sides of the equation, since the bases are the same.
Solve for x: Solve for $x$ by isolating it on one side of the equation using basic algebraic manipulations.

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

A: Here are some common mistakes to avoid when solving exponential equations:

Forgetting to equate exponents: When the bases are the same, the exponents must be equal. Don't forget to equate the exponents on both sides of the equation.
Not using the properties of exponents: Exponents have several useful properties that can help simplify complex expressions. Don't be afraid to use them to your advantage.
Not isolating the variable: To solve for the variable, isolate it on one side of the equation using basic algebraic manipulations.

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

A: Exponential equations have numerous real-world applications in fields such as finance, economics, and computer science. Here are a few examples:

Compound interest: Exponential equations are used to calculate compound interest in finance. For example, if you deposit $1000 into a savings account with a 5% annual interest rate, the balance after one year will be $1050, after two years will be $1102.50, and so on.
Population growth: Exponential equations are used to model population growth in biology and ecology. For example, if a population grows at a rate of 2% per year, the population after one year will be 102% of the original population, after two years will be 104.04% of the original population, and so on.
Computer science: Exponential equations are used in computer science to model the growth of algorithms and data structures. For example, if an algorithm has a time complexity of O(n^2), the running time will grow exponentially with the size of the input.

Q: How do I determine the base of an exponential equation?

A: To determine the base of an exponential equation, look for the number that is being raised to a power. For example, in the equation $2^x = 8$ , the base is 2.

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

A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power. A linear equation, on the other hand, is an equation that involves a linear expression, which is a number multiplied by a variable. For example, the equation $2x = 4$ is a linear equation because it involves a linear expression $2x$ , while the equation $2^x = 8$ is an exponential equation because it involves an exponential expression $2^x$ .

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

A: Yes, you can use a calculator to solve exponential equations. However, it's essential to understand the underlying principles of exponential equations and how to solve them using algebraic manipulations and properties of exponents. A calculator can be a useful tool for checking your work and verifying your solutions.

Q: How do I graph an exponential equation?

A: To graph an exponential equation, you can use a graphing calculator or a computer program. Here are the general steps:

Enter the equation: Enter the exponential equation into the graphing calculator or computer program.
Set the window: Set the window to display the graph of the equation.
Graph the equation: Graph the equation using the graphing calculator or computer program.

By following these steps, you can create a graph of the exponential equation and visualize its behavior.