Solving The Exponential Equation 2^x + 3^(x-2) = 3^x - 2^(x+1)

Introduction: The Intriguing Equation

In the realm of mathematical challenges, exponential equations hold a special allure. They beckon us to unravel the mysteries hidden within the interplay of exponents and variables. Today, we embark on a journey to dissect and solve the intriguing equation 2^x + 3^(x-2) = 3^x - 2^(x+1). This equation, seemingly simple at first glance, presents a fascinating puzzle that requires careful manipulation and a touch of algebraic finesse. Our exploration will not only unveil the solution but also illuminate the underlying principles that govern exponential equations. Let's delve into the heart of this equation and discover the elegant solution it holds.

Reorganizing the Equation: A Strategic Maneuver

Our initial step in deciphering the equation 2^x + 3^(x-2) = 3^x - 2^(x+1) involves a strategic rearrangement of terms. The goal here is to group similar terms together, paving the way for simplification. We begin by adding 2^(x+1) to both sides of the equation. This move elegantly relocates the term from the right-hand side to the left, setting the stage for combining terms with the same base. Similarly, we subtract 3^(x-2) from both sides, effectively moving this term from the left-hand side to the right. This strategic dance of terms yields a transformed equation that is more amenable to analysis.

The equation now takes on a new form: 2^x + 2^(x+1) = 3^x - 3^(x-2). This rearrangement is not merely cosmetic; it serves a crucial purpose. By grouping terms with the same base (2 on the left and 3 on the right), we create an opportunity to factor out common factors. Factoring is a powerful technique in algebra, allowing us to simplify expressions and reveal hidden structures. In our case, factoring will lead us closer to isolating the variable x and ultimately uncovering the solution.

This strategic maneuver of reorganizing terms highlights a fundamental principle in problem-solving: transforming a problem into a more manageable form. By carefully manipulating the equation, we have created a new perspective that brings us closer to the solution. The next step involves leveraging the power of factoring to further simplify the equation and unveil the value of x.

Factoring and Simplifying: Unveiling the Structure

Having strategically rearranged our equation to 2^x + 2^(x+1) = 3^x - 3^(x-2), we now harness the power of factoring to simplify the expressions on both sides. On the left-hand side, we observe that 2^x is a common factor. By factoring out 2^x, we transform the left-hand side into 2^x(1 + 2). This factorization elegantly collapses two terms into a single, more compact expression. The term (1 + 2) simplifies to 3, further streamlining the equation.

On the right-hand side, the common factor is 3^(x-2). Factoring this out, we obtain 3^(x-2)(32 - 1). Here, we've used the property that 3^x = 3^(x-2) * 3^2 to facilitate the factorization. The expression (3^2 - 1) simplifies to (9 - 1), which equals 8. This simplification further reduces the complexity of the equation.

With the factoring complete, our equation now stands as 2^x * 3 = 3^(x-2) * 8. This simplified form is a testament to the power of factoring. We have successfully transformed the original equation into a more transparent structure, where the relationship between the terms is clearer. The next step involves isolating the exponential terms and preparing the equation for further manipulation. This meticulous simplification brings us closer to unveiling the solution for x.

Isolating Exponential Terms: A Crucial Step

Following our successful factoring and simplification, the equation now reads 2^x * 3 = 3^(x-2) * 8. Our next strategic move involves isolating the exponential terms. This means grouping the terms with the variable x on one side of the equation and the constant terms on the other. To achieve this, we divide both sides of the equation by 3. This action effectively moves the constant factor 3 from the left-hand side to the right-hand side. Simultaneously, we divide both sides by 3^(x-2), relocating this exponential term from the right-hand side to the left.

This careful manipulation results in a new form of the equation: 2^x / 3^(x-2) = 8 / 3. This rearrangement is a pivotal step in our solution journey. By isolating the exponential terms, we have created an opportunity to combine them into a single expression. This consolidation will allow us to leverage the properties of exponents and further simplify the equation.

The isolated exponential terms now stand poised for combination. The next step involves skillfully applying the rules of exponents to merge these terms into a more manageable form. This process will bring us closer to expressing the equation in a way that allows us to directly solve for x. The isolation of exponential terms is a testament to our strategic approach, guiding us towards the solution with each deliberate step.

Combining Exponential Terms: Harnessing the Power of Exponents

With the exponential terms isolated in the equation 2^x / 3^(x-2) = 8 / 3, we now embark on the crucial step of combining them. To achieve this, we employ the fundamental properties of exponents. Recall that dividing terms with exponents can be simplified by subtracting the exponents. However, our terms have different bases (2 and 3), so we need a slightly different approach.

First, let's rewrite 3^(x-2) as 3^x / 3^2. This manipulation allows us to express the left-hand side of the equation as 2^x / (3^x / 3^2). Now, dividing by a fraction is the same as multiplying by its reciprocal, so we can rewrite this as 2^x * (3^2 / 3^x). This transformation brings the terms with the same exponent, x, closer together.

Next, we rearrange the terms to group the exponents together: (2^x / 3^x) * 3^2 = 8 / 3. Now we can use the property that a^x / b^x = (a/b)^x. Applying this rule, we get (2/3)^x * 3^2 = 8 / 3. Since 3^2 = 9, our equation now reads (2/3)^x * 9 = 8 / 3. This is a significant step forward, as we have successfully combined the exponential terms into a single expression.

The equation is now in a form where we can isolate the exponential term and prepare it for further manipulation. The strategic application of exponent properties has allowed us to consolidate the terms, bringing us closer to solving for x. The next step involves isolating the (2/3)^x term and then employing logarithms to unveil the value of x.

Introducing Logarithms: Unlocking the Value of x

Having simplified our equation to (2/3)^x * 9 = 8 / 3, we now face the task of isolating x, which resides within the exponent. To liberate x from its exponential confinement, we introduce the powerful tool of logarithms. Before we directly apply logarithms, we need to isolate the exponential term. We achieve this by dividing both sides of the equation by 9, yielding (2/3)^x = 8 / (3 * 9), which simplifies to (2/3)^x = 8 / 27.

Now, the equation is poised for the application of logarithms. Logarithms are the inverse operation of exponentiation, allowing us to