Solving For X In 3^(2x+1)=3^(x+5) Exponential Equation
Introduction
In the realm of mathematics, solving exponential equations is a fundamental skill. These equations involve variables in the exponents, and finding the value of the variable requires a solid understanding of exponential properties. In this article, we will delve into the process of solving the exponential equation 3^(2x+1) = 3^(x+5), and we will methodically navigate the steps to determine the value of x. This example serves as an excellent illustration of how to tackle exponential equations where the bases are the same. By equating the exponents, we can transform the problem into a simple algebraic equation, leading us to the solution. Whether you are a student grappling with exponential equations or a math enthusiast looking to reinforce your skills, this detailed explanation will provide you with a clear and concise approach to solving such problems.
Understanding Exponential Equations
Before we dive into the specific problem, let's take a moment to understand the fundamental principles of exponential equations. An exponential equation is an equation in which the variable appears 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. Solving exponential equations involves finding the value of x that satisfies the equation. One of the key properties we use when solving exponential equations is that if we have two exponential expressions with the same base that are equal to each other, then their exponents must be equal. Mathematically, this can be expressed as follows: If a^m = a^n, then m = n. This property is the cornerstone of solving many exponential equations, as it allows us to transform a potentially complex exponential problem into a simpler algebraic one. Understanding this principle is crucial for tackling equations like 3^(2x+1) = 3^(x+5), where we can directly equate the exponents once we recognize that the bases are the same.
Problem Statement: 3^(2x+1) = 3^(x+5)
Now, let's formally state the problem we aim to solve. We are given the exponential equation 3^(2x+1) = 3^(x+5). Our goal is to find the value of x that makes this equation true. This equation features two exponential expressions with the same base, which is 3. The exponents are 2x + 1 on the left side and x + 5 on the right side. The equal sign indicates that these two exponential expressions have the same value. To solve for x, we will utilize the property that states if the bases are equal, then the exponents must be equal. This will allow us to set up a linear equation by equating the exponents and then solving for x using basic algebraic techniques. This problem is a classic example of how to apply the fundamental principles of exponential equations to find the unknown variable. By carefully following the steps, we can determine the specific value of x that satisfies the given equation.
Step-by-Step Solution
To solve the exponential equation 3^(2x+1) = 3^(x+5), we will follow a step-by-step approach. This will ensure clarity and accuracy in our solution.
Step 1: Equate the Exponents
The first and most crucial step is to recognize that the bases on both sides of the equation are the same (both are 3). As we discussed earlier, if the bases are equal, then the exponents must be equal. Therefore, we can set the exponents equal to each other:
2x + 1 = x + 5
This step transforms the exponential equation into a linear equation, which is much easier to solve. We have successfully eliminated the exponential terms and now have a simple algebraic equation to work with.
Step 2: Isolate the Variable
Now that we have the linear equation 2x + 1 = x + 5, we need to isolate the variable x. To do this, we will perform algebraic manipulations to get all the x terms on one side of the equation and the constant terms on the other side. First, let's subtract x from both sides of the equation:
2x + 1 - x = x + 5 - x
This simplifies to:
x + 1 = 5
Next, we subtract 1 from both sides of the equation to isolate x:
x + 1 - 1 = 5 - 1
This simplifies to:
x = 4
Step 3: Verify the Solution
It's always a good practice to verify our solution by plugging the value of x back into the original equation. This ensures that our solution is correct and that we haven't made any mistakes in our calculations. Substitute x = 4 into the original equation 3^(2x+1) = 3^(x+5):
3^(2(4)+1) = 3^(4+5)
Simplify the exponents:
3^(8+1) = 3^(9)
3^9 = 3^9
Since both sides of the equation are equal, our solution x = 4 is correct. This verification step confirms that the value we found for x indeed satisfies the original exponential equation.
Answer
Therefore, the value of x that satisfies the equation 3^(2x+1) = 3^(x+5) is 4. So the correct answer is C. 4.
Conclusion
In this article, we successfully solved the exponential equation 3^(2x+1) = 3^(x+5) by applying the fundamental principles of exponential equations. We began by understanding the basic concepts of exponential equations and the key property that allows us to equate exponents when the bases are the same. We then methodically worked through the problem, equating the exponents, isolating the variable, and verifying our solution. The step-by-step approach we followed ensures clarity and accuracy in our solution. We found that the value of x that satisfies the equation is 4. This problem serves as an excellent example of how to tackle exponential equations where the bases are the same. By mastering these techniques, you can confidently solve a wide range of exponential equations. Remember to always verify your solution to ensure accuracy and to reinforce your understanding of the concepts involved. Exponential equations are a crucial part of mathematics, and a solid grasp of these concepts will be beneficial in various mathematical contexts.
