Solving The Exponential Equation 9^x - 1 = 2 A Step-by-Step Guide

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Hey math enthusiasts! Ever stumbled upon an equation that looks a bit intimidating at first glance? Well, today we're going to tackle one such equation together: 9^x - 1 = 2. Don't worry, we'll break it down step by step, making sure everyone can follow along. Our mission? To find the value of 'x' that makes this equation true. So, buckle up, grab your thinking caps, and let's dive into the fascinating world of exponents and equations!

Decoding the Equation: 9^x - 1 = 2

When we first see an equation like 9^x - 1 = 2, it might seem a little daunting, especially with that 'x' up there in the exponent. But fear not! The beauty of mathematics lies in its systematic approach to problem-solving. To solve for 'x', our ultimate goal is to isolate it on one side of the equation. This means we need to carefully undo the operations that are affecting 'x'. Think of it like peeling away layers of an onion – each step brings us closer to the core, which in this case, is the value of 'x'.

Let's start by taking a closer look at the equation itself. We have 9 raised to the power of 'x', then we subtract 1, and the result is 2. The first thing we want to do is get rid of that '- 1'. How do we do that? By performing the opposite operation: adding 1 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to maintain the balance. This is a fundamental principle in algebra, ensuring that the equation remains true throughout our manipulations.

So, adding 1 to both sides gives us:

9^x - 1 + 1 = 2 + 1

This simplifies to:

9^x = 3

Now, we're one step closer! We've isolated the term with 'x' in the exponent. But how do we get that 'x' down from the exponent? This is where our knowledge of exponents and logarithms comes into play. We need to express both sides of the equation with the same base. Why? Because if we have the same base raised to different powers, and they are equal, then the powers themselves must be equal. This is a crucial concept for solving exponential equations.

Can we express both 9 and 3 with the same base? Absolutely! We know that 9 is 3 squared (3^2). So, we can rewrite the equation as:

(32)x = 3

Now, we have a power raised to another power. Remember the rule of exponents that says (am)n = a^(m*n)? We can apply that here:

3^(2x) = 3

Ah, we're getting there! Now, we have the same base on both sides. But what's the exponent on the right side? Well, any number raised to the power of 1 is itself. So, we can rewrite 3 as 3^1:

3^(2x) = 3^1

Now, the magic happens! Since the bases are the same, we can equate the exponents:

2x = 1

And finally, to isolate 'x', we divide both sides by 2:

x = 1/2

So, there we have it! The solution to the equation 9^x - 1 = 2 is x = 1/2. We've successfully navigated the world of exponents and equations, using our knowledge of algebraic manipulations and exponent rules to find the answer. Wasn't that a rewarding journey?

A Deeper Dive: Why x = 1/2 Works

Now that we've found the solution, it's always a good idea to take a moment and verify our answer. This not only gives us confidence in our solution but also deepens our understanding of the equation itself. So, let's plug x = 1/2 back into the original equation: 9^x - 1 = 2.

Substituting x = 1/2, we get:

9^(1/2) - 1 = 2

But what does 9^(1/2) mean? Remember that a fractional exponent represents a root. Specifically, a number raised to the power of 1/2 is the same as taking the square root of that number. So, 9^(1/2) is the square root of 9.

The square root of 9 is 3 (since 3 * 3 = 9). Therefore, our equation becomes:

3 - 1 = 2

And this is indeed true! 2 = 2. This confirms that our solution, x = 1/2, is correct. We've not only found the answer but also verified it, solidifying our understanding of the problem.

But let's take this a step further. Why does a fractional exponent represent a root? This is a fundamental concept in exponents, and understanding it will help us tackle more complex problems in the future. Think about it this way: we know that x^2 * x^2 = x^(2+2) = x^4. This is the product of powers rule: when multiplying exponents with the same base, we add the powers.

Now, what if we wanted to find a value that, when multiplied by itself, gives us x? That's the square root of x, which we can write as √x. Let's represent √x as x^n, where 'n' is some exponent we need to find. So, we have:

x^n * x^n = x

Using the product of powers rule, we get:

x^(n+n) = x^1

x^(2n) = x^1

Since the bases are the same, we can equate the exponents:

2n = 1

Dividing both sides by 2, we get:

n = 1/2

Therefore, √x = x^(1/2). This elegant demonstration shows why a fractional exponent of 1/2 represents the square root. The same logic can be extended to other fractional exponents. For example, x^(1/3) represents the cube root of x, and so on.

By understanding the underlying principles of exponents and roots, we can confidently tackle a wide range of mathematical problems. The equation 9^x - 1 = 2 might have seemed a bit tricky at first, but by breaking it down step by step and applying our knowledge of mathematical rules, we were able to find the solution and verify it. And that, my friends, is the power of mathematics!

Exploring Alternative Approaches to Solve 9^x - 1 = 2

While we've successfully solved the equation 9^x - 1 = 2 by expressing both sides with the same base, it's always beneficial to explore alternative approaches. This not only expands our problem-solving toolkit but also provides a deeper appreciation for the interconnectedness of mathematical concepts. So, let's consider another method: using logarithms.

Logarithms are the inverse operation of exponentiation. Just as subtraction is the inverse of addition and division is the inverse of multiplication, logarithms