Solving The Exponential Equation 3^(x+1) - 2 * 3^x - 9 = 0

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This article delves into the step-by-step solution of the exponential equation 3^(x+1) - 2 * 3^x - 9 = 0. Exponential equations, where the variable appears in the exponent, often require clever algebraic manipulation and a solid understanding of exponent rules. This comprehensive guide will not only provide the solution but also illuminate the underlying concepts and techniques applicable to a broader range of exponential equations.

Understanding Exponential Equations

Before diving into the solution, it's crucial to grasp the fundamental principles of exponential equations. An exponential equation is an equation in which the variable appears in the exponent. These equations arise in various fields, including finance (compound interest), biology (population growth), and physics (radioactive decay). Solving them often involves isolating the exponential term and then employing logarithms or other techniques to extract the variable from the exponent. Key properties of exponents, such as a^(m+n) = a^m * a^n and a^(m*n) = (am)n, play a vital role in simplifying and solving these equations.

Rewriting the Equation

The first step in solving 3^(x+1) - 2 * 3^x - 9 = 0 is to rewrite the equation using exponent rules. Specifically, we can rewrite 3^(x+1) as 3^x * 3^1, which simplifies to 3 * 3^x. This transformation allows us to group the terms containing 3^x and factor them out. Our equation now becomes:

3 * 3^x - 2 * 3^x - 9 = 0

This step is crucial because it consolidates the exponential terms, making the equation easier to manipulate. By recognizing and applying the exponent rule, we've set the stage for further simplification.

Factoring Out the Exponential Term

Now, we can factor out the common term, 3^x, from the first two terms of the equation:

3^x * (3 - 2) - 9 = 0

This simplifies to:

3^x * 1 - 9 = 0

Further simplifying, we get:

3^x - 9 = 0

Factoring is a fundamental algebraic technique that allows us to isolate the exponential term. By recognizing the common factor, we've reduced the equation to a simpler form that is easier to solve.

Isolating the Exponential Term

The next step is to isolate the exponential term, 3^x. We can do this by adding 9 to both sides of the equation:

3^x = 9

Isolating the exponential term is a critical step in solving exponential equations. It allows us to directly relate the exponential expression to a constant value, paving the way for the application of logarithms or other techniques to solve for the variable.

Expressing Both Sides with the Same Base

To solve for x, we need to express both sides of the equation with the same base. We know that 9 can be written as 3^2. Therefore, we can rewrite the equation as:

3^x = 3^2

Expressing both sides with the same base is a powerful technique for solving exponential equations. When the bases are the same, we can equate the exponents, transforming the exponential equation into a simple algebraic equation.

Equating the Exponents

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

x = 2

This step is the culmination of our efforts. By expressing both sides of the equation with the same base, we've transformed the exponential equation into a simple algebraic equation that can be solved directly.

Verifying the Solution

It's always a good practice to verify the solution by substituting it back into the original equation:

3^(2+1) - 2 * 3^2 - 9 = 0

3^3 - 2 * 9 - 9 = 0

27 - 18 - 9 = 0

0 = 0

Since the equation holds true, our solution x = 2 is correct. Verification is a crucial step in problem-solving, as it ensures that our solution satisfies the original equation and that no errors were made during the solution process.

Alternative Methods for Solving Exponential Equations

While we solved this equation by expressing both sides with the same base, another common technique involves using logarithms. Let's explore how we could have solved the equation using logarithms.

Using Logarithms

After isolating the exponential term (3^x = 9), we could have taken the logarithm of both sides. Using the natural logarithm (ln) or the common logarithm (log base 10) would work:

ln(3^x) = ln(9)

Using the power rule of logarithms, which states that ln(a^b) = b * ln(a), we can rewrite the equation as:

x * ln(3) = ln(9)

Now, we can solve for x by dividing both sides by ln(3):

x = ln(9) / ln(3)

Since 9 = 3^2, we can rewrite ln(9) as ln(3^2). Again using the power rule of logarithms:

x = (2 * ln(3)) / ln(3)

Simplifying, we get:

x = 2

As we can see, using logarithms leads to the same solution, x = 2. This alternative method highlights the versatility of logarithms in solving exponential equations.

When to Use Logarithms

Logarithms are particularly useful when it's not straightforward to express both sides of the equation with the same base. For example, consider the equation 5^x = 12. In this case, it's not easy to express 12 as a power of 5. Therefore, taking the logarithm of both sides would be the most efficient approach to solve for x.

Common Mistakes to Avoid

When solving exponential equations, it's essential to avoid common mistakes that can lead to incorrect solutions. Here are some pitfalls to watch out for:

  • Incorrectly Applying Exponent Rules: Ensure that you correctly apply exponent rules such as a^(m+n) = a^m * a^n and a^(m*n) = (am)n. A common mistake is to incorrectly distribute exponents or combine terms.
  • Forgetting to Verify the Solution: Always verify your solution by substituting it back into the original equation. This step helps to catch any errors made during the solution process.
  • Not Isolating the Exponential Term: Before applying logarithms or other techniques, make sure to isolate the exponential term. This simplifies the equation and makes it easier to solve.
  • Dividing by Zero: Be cautious when dividing both sides of the equation by an expression containing the variable. Ensure that the expression is not equal to zero.
  • Ignoring Extraneous Solutions: In some cases, the algebraic manipulations may introduce extraneous solutions that do not satisfy the original equation. Therefore, it's crucial to verify all solutions.

Practice Problems

To solidify your understanding of solving exponential equations, try these practice problems:

  1. Solve for x: 2^(x+2) - 3 * 2^x = 16
  2. Solve for x: 4^x - 2^(x+1) - 8 = 0
  3. Solve for x: 9^x - 3^(x+1) = 54

Working through these problems will help you develop your skills and confidence in solving exponential equations.

Conclusion

Solving the exponential equation 3^(x+1) - 2 * 3^x - 9 = 0 demonstrates the importance of algebraic manipulation, exponent rules, and the strategic use of logarithms. By rewriting the equation, factoring out common terms, and expressing both sides with the same base, we arrived at the solution x = 2. We also explored an alternative method using logarithms, highlighting the versatility of this technique. Avoiding common mistakes and practicing with various problems are key to mastering exponential equations. Understanding these concepts and techniques will not only help you solve similar equations but also provide a solid foundation for more advanced mathematical concepts. Remember to always verify your solution and practice consistently to improve your skills. Solving exponential equations can be challenging, but with a methodical approach and a strong grasp of the fundamentals, you can confidently tackle these problems. Mastering exponential equations is a valuable skill in mathematics and its applications. Keep practicing and you'll become proficient in solving them.