Solve Exponential Equations: Find The Value Of X

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Hey guys! Today, we're diving into a super common type of problem you'll see in algebra: solving exponential equations where the bases are the same. Specifically, we're tackling the equation 3^(2x+1) = 3^(x+5). Our mission? To find the value of 'x' that makes this equation true. Don't worry, it's easier than it looks! Let's break it down step-by-step.

Understanding the Basics of Exponential Equations

Before we jump into solving, let's quickly recap what exponential equations are all about. An exponential equation is an equation in which the variable appears in the exponent. The key to solving many exponential equations lies in understanding that if you have the same base on both sides of the equation, then the exponents must be equal. This is a fundamental property that allows us to simplify and solve for the variable.

In our case, we have 3^(2x+1) = 3^(x+5). Notice that both sides of the equation have the same base, which is 3. This is excellent news because it means we can use that property we just talked about! When the bases are the same, we can equate the exponents. This transforms the exponential equation into a simple algebraic equation that we can solve using basic algebraic techniques.

So, the main idea here is to recognize when you can equate exponents. This usually happens when you can express both sides of the equation with the same base. Sometimes, it might require a bit of algebraic manipulation to get the bases to match, but once they do, you're golden! Keep this in mind as we proceed with solving our specific equation. This is a very useful skill in math, especially in algebra and calculus, so make sure to grasp the concept.

Step-by-Step Solution

Okay, let's get our hands dirty and solve the equation 3^(2x+1) = 3^(x+5). Remember, the first crucial step is recognizing that the bases are the same. Since both sides of the equation have a base of 3, we can confidently equate the exponents. This means we can set the exponent on the left side equal to the exponent on the right side:

2x + 1 = x + 5

Now we have a simple linear equation to solve for 'x'. Our next goal is to isolate 'x' on one side of the equation. To do this, let's subtract 'x' from both sides:

2x - x + 1 = x - x + 5

This simplifies to:

x + 1 = 5

Now, to completely isolate 'x', we need to get rid of the '+1' on the left side. We can do this by subtracting 1 from both sides of the equation:

x + 1 - 1 = 5 - 1

Which simplifies to:

x = 4

And that's it! We've found the value of 'x' that satisfies the original equation. The solution is x = 4. To be absolutely sure, we can plug this value back into the original equation to check our work. This is always a good practice to ensure you haven't made any mistakes along the way. Let's do that in the next section.

Verification of the Solution

Alright, let's verify that x = 4 is indeed the correct solution to the equation 3^(2x+1) = 3^(x+5). To do this, we'll substitute '4' for 'x' in both sides of the original equation and see if they are equal. This is a crucial step to confirm that our solution is accurate.

First, let's substitute 'x = 4' into the left side of the equation, which is 3^(2x+1):

3^(2(4)+1) = 3^(8+1) = 3^9

Now, let's substitute 'x = 4' into the right side of the equation, which is 3^(x+5):

3^(4+5) = 3^9

As you can see, both sides of the equation simplify to 3^9 when we substitute 'x = 4'. This confirms that our solution is correct! Verification is a key part of problem-solving, especially in mathematics. It helps prevent errors and gives you confidence in your answer. So, always remember to check your solutions whenever possible.

Therefore, we can confidently say that the value of 'x' that satisfies the equation 3^(2x+1) = 3^(x+5) is indeed 4.

Alternative Approaches

While equating exponents is the most straightforward method for solving this particular equation, it's worth mentioning some alternative approaches that might be useful in different scenarios. These approaches can provide you with more tools in your problem-solving toolkit.

Logarithmic Approach

One alternative is to use logarithms. If you have an equation like 3^(2x+1) = 3^(x+5), you could take the logarithm of both sides. Remember, whatever you do to one side of the equation, you must do to the other to maintain equality. You can use any base for the logarithm, but the common logarithm (base 10) or the natural logarithm (base 'e') are often convenient.

Taking the natural logarithm of both sides, we get:

ln(3^(2x+1)) = ln(3^(x+5))

Using the property of logarithms that allows you to bring the exponent down as a coefficient, we have:

(2x+1)ln(3) = (x+5)ln(3)

Now, you can divide both sides by ln(3) since it's a non-zero constant:

2x + 1 = x + 5

And from here, you would solve for 'x' as we did before, arriving at x = 4.

Graphical Approach

Another approach, especially useful for visualizing the solution, is to graph the two functions y = 3^(2x+1) and y = 3^(x+5). The point where the two graphs intersect represents the solution to the equation. While this method might not give you an exact answer without the aid of technology (like a graphing calculator or software), it can give you a good approximation and a visual understanding of the problem.

Using a graphing tool, you would plot both equations and find the x-coordinate of their intersection point. You'll see that the graphs intersect at x = 4.

These alternative approaches might be more useful or necessary when dealing with more complex exponential equations where you can't easily equate the exponents. Knowing these methods broadens your problem-solving skills and helps you tackle a wider range of problems.

Practice Problems

To really solidify your understanding, let's try a few practice problems. Working through these will help you become more confident in solving exponential equations. Remember, practice makes perfect!

  1. Solve for x: 5^(3x-2) = 5^(x+4)
  2. Solve for x: 2^(4x+1) = 2^(2x+7)
  3. Solve for x: 4^(x+2) = 4^(3x-1)

Try to solve these problems using the method we discussed earlier: equating the exponents. And don't forget to verify your solutions! The answers are provided below, but try to solve them on your own first.

Solutions to Practice Problems:

  1. x = 3
  2. x = 3
  3. x = 3/2 or 1.5

If you got these right, awesome job! If not, go back and review the steps, and try again. Understanding the process is key.

Conclusion

So, there you have it! Solving the equation 3^(2x+1) = 3^(x+5) is all about recognizing the equal bases and equating the exponents. Remember, this technique is widely applicable to various exponential equations, and mastering it will definitely boost your algebra skills. Keep practicing, and you'll become a pro at solving these types of problems in no time! Keep an eye out for more math tips and tricks, and happy solving!