Equivalent Equation To 9^(x-3) = 729? Solve It Now!
Hey guys! Today, we're diving into a super interesting math problem that involves exponents and equations. We're going to figure out which equation is equivalent to 9^(x-3) = 729. This might seem a bit daunting at first, but trust me, we'll break it down step-by-step so it's super easy to understand. So, grab your thinking caps, and let's get started!
Understanding the Problem
Before we jump into solving, let's make sure we understand what the question is asking. We have the equation 9^(x-3) = 729, and we need to find another equation that has the same solution for x. In other words, we're looking for an equivalent equation. This usually involves rewriting one or both sides of the equation in a different form, while maintaining the equality. The key here is to recognize that 729 can be expressed as a power of 9, which will help us simplify things. Let's explore how we can do that.
Powers and Exponents
To solve this, we need to understand powers and exponents. Remember, an exponent tells you how many times to multiply a number (the base) by itself. For example, 9^2 means 9 multiplied by itself (9 * 9), which equals 81. Similarly, 9^3 means 9 * 9 * 9, and so on. The goal here is to express both sides of the equation using the same base, which will allow us to compare the exponents directly. Now, let’s see how 729 fits into this picture. Can we express 729 as a power of 9? This is the critical step in simplifying our equation and finding the equivalent form. Once we figure this out, the rest becomes much easier.
Rewriting 729 as a Power of 9
This is the crucial part! We need to figure out how to express 729 as a power of 9. Let's try it out: 9^1 = 9, 9^2 = 81, and 9^3 = 729. Bingo! We've found that 729 is equal to 9^3. This is a significant breakthrough because now we can rewrite our original equation using the same base on both sides. This makes comparing the exponents directly much simpler. By expressing 729 as 9^3, we've taken the first big step towards finding an equivalent equation. Keep this in mind as we move forward – the ability to rewrite numbers using exponents is a fundamental skill in solving these types of problems.
Finding the Equivalent Equation
Now that we know 729 is the same as 9^3, we can rewrite the original equation. Our equation 9^(x-3) = 729 becomes 9^(x-3) = 9^3. See how much simpler that looks? Now we have the same base (9) on both sides of the equation. This is fantastic because it allows us to focus solely on the exponents. If the bases are the same, then for the equation to hold true, the exponents must also be equal. So, we can set the exponents equal to each other and solve for x. This is a common technique when dealing with exponential equations, and it makes the problem much more manageable.
Equating the Exponents
Since the bases are the same (both are 9), we can equate the exponents. This means we can set the exponent on the left side of the equation, which is (x-3), equal to the exponent on the right side, which is 3. So we get a new equation: x - 3 = 3. This is a simple linear equation that we can easily solve for x. This step is a direct consequence of the property that if a^m = a^n, then m = n, provided that a is not 0, 1, or -1. In our case, a is 9, so this property applies perfectly. Now, let’s solve this linear equation and find the value of x.
Solving for x
To solve the equation x - 3 = 3, we need to isolate x. We can do this by adding 3 to both sides of the equation. This gives us x - 3 + 3 = 3 + 3, which simplifies to x = 6. So, the solution to our equation is x = 6. This tells us that the value of x that makes the original equation true is 6. Now that we’ve found x, we can go back to the original question and determine which of the given options is an equivalent equation. Remember, an equivalent equation will also be true when x = 6.
Analyzing the Answer Choices
Okay, we've found that 9^(x-3) = 729 is equivalent to 9^(x-3) = 9^3. Now let's look at the answer choices and see which one matches. This is where we put all our hard work to the test! We need to compare our simplified equation with the given options to see which one is the same. This involves careful observation and understanding of what we've already accomplished. Remember, the key is to identify the equation that represents the same relationship as 9^(x-3) = 9^3. Let’s go through each option one by one.
Evaluating Each Option
Let's go through each option to see which one is equivalent to 9^(x-3) = 729:
- A. 9^(x-3) = 9^81: This is incorrect. We know that 729 is 9^3, not 9^81.
- B. 9^(x-3) = 9^3: This is the correct answer! We found that 729 is equal to 9^3, so this equation is equivalent to the original.
- C. 3^(x-3) = 3^6: Let's analyze this one. If we rewrite the original equation in base 3, we get (32)(x-3) = 3^6, which simplifies to 3^(2x-6) = 3^6. This is not the same as option C.
- D. 3^(2x-3) = 3^6: This is also incorrect. As we saw in the analysis of option C, the equivalent equation in base 3 should be 3^(2x-6) = 3^6, not 3^(2x-3) = 3^6.
So, the correct answer is B. 9^(x-3) = 9^3.
Final Answer
Alright, guys! After breaking down the problem step-by-step, we've nailed it! The equation equivalent to 9^(x-3) = 729 is indeed B. 9^(x-3) = 9^3. We started by understanding the problem, rewriting 729 as a power of 9, equating the exponents, and finally, analyzing the answer choices. This methodical approach helped us solve the problem efficiently and accurately. Remember, the key to solving these types of problems is to break them down into smaller, manageable steps. With practice, you'll become a pro at solving exponential equations! Keep up the great work, and I'll see you in the next math adventure!