Solving Exponential Equations: 3*(9^x) - 10*(3^x) + 3 = 0

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Solving Exponential Equations: 3*(9^x) - 10*(3^x) + 3 = 0

Hey guys! Today, we're diving into a fun math problem: solving the exponential equation 3*(9^x) - 10*(3^x) + 3 = 0. Exponential equations might seem intimidating at first, but with the right approach, they can be broken down into manageable steps. So, let's grab our mathematical tools and get started!

Understanding Exponential Equations

Before we jump into solving this specific equation, let's quickly recap what exponential equations are. In essence, exponential equations are equations where the variable appears in the exponent. These equations often describe phenomena that grow or decay exponentially, such as population growth, radioactive decay, and compound interest. Understanding their properties is crucial for tackling real-world problems.

Key characteristics of exponential equations include:

  • The variable (usually denoted as 'x') is part of the exponent.
  • The base is a constant (in our case, 9 and 3).
  • They can represent rapid growth or decay depending on the base and exponent.

When faced with an exponential equation, our goal is to isolate the exponential term and then use logarithms or other techniques to solve for the variable. Now that we have a basic understanding, let’s apply these concepts to our given equation.

Breaking Down the Equation: 3*(9^x) - 10*(3^x) + 3 = 0

Okay, let's look closely at our equation: 3*(9^x) - 10*(3^x) + 3 = 0. At first glance, it might appear complex, but we can simplify it using a clever substitution. This is a common strategy for solving many types of equations.

The trick here is to recognize that 9^x can be rewritten as (32)x, which is the same as (3x)2. This allows us to make a substitution that transforms the equation into a more familiar quadratic form. By doing this, we are essentially changing the structure of the equation without changing its solution. This is a powerful technique in algebra and calculus, making seemingly intractable problems solvable.

The Substitution Technique

Let's substitute y = 3^x. This simplifies our equation significantly. Replacing every instance of 3^x with y, and (3x)2 with y^2, our equation becomes:

3y^2 - 10y + 3 = 0

See? Much simpler! This is now a quadratic equation, which we know how to solve. Quadratic equations are a cornerstone of algebra, and mastering them opens doors to solving a wide range of problems. This transformation highlights a key aspect of mathematical problem-solving: reducing complex problems into simpler, manageable forms.

Solving the Quadratic Equation

Now we have the quadratic equation 3y^2 - 10y + 3 = 0. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. For this particular equation, factoring is the most straightforward approach.

Factoring the Quadratic

We need to find two numbers that multiply to (3 * 3 = 9) and add up to -10. Those numbers are -9 and -1. So, we can rewrite our quadratic equation as:

3y^2 - 9y - y + 3 = 0

Now, we factor by grouping:

3y(y - 3) - 1(y - 3) = 0

(3y - 1)(y - 3) = 0

Finding the Solutions for y

Setting each factor equal to zero gives us two possible solutions for y:

3y - 1 = 0 => y = 1/3

y - 3 = 0 => y = 3

So, we have two values for y: 1/3 and 3. But remember, we're not trying to solve for y; we need to find the values of x. This is a critical step in solving equations after a substitution: always remember to substitute back to find the original variable's value.

Substituting Back to Find x

We recall our earlier substitution: y = 3^x. Now we need to substitute our values of y back into this equation to find the corresponding values of x.

Case 1: y = 1/3

Substituting y = 1/3, we have:

3^x = 1/3

We can rewrite 1/3 as 3^(-1), so:

3^x = 3^(-1)

Since the bases are equal, the exponents must be equal. Therefore:

x = -1

Case 2: y = 3

Substituting y = 3, we have:

3^x = 3

Here, 3 is the same as 3^1, so:

3^x = 3^1

Again, since the bases are equal, the exponents must be equal:

x = 1

The Solutions for x

So, we've found two solutions for x: x = -1 and x = 1. These are the values that satisfy the original exponential equation. It's always a good idea to check these solutions by plugging them back into the original equation to ensure they are correct.

Verifying the Solutions

Let's verify our solutions by substituting them back into the original equation: 3*(9^x) - 10*(3^x) + 3 = 0.

For x = -1:

3*(9^(-1)) - 10*(3^(-1)) + 3 = 3*(1/9) - 10*(1/3) + 3 = 1/3 - 10/3 + 3 = (1 - 10 + 9)/3 = 0

So, x = -1 is indeed a solution.

For x = 1:

3*(9^(1)) - 10*(3^(1)) + 3 = 39 - 103 + 3 = 27 - 30 + 3 = 0

And x = 1 is also a solution. Woo-hoo! We got them both right.

Key Takeaways

Let's recap the key steps we took to solve this exponential equation. This will help solidify our understanding and provide a clear strategy for tackling similar problems in the future. Remember, practice makes perfect, so the more you solve, the better you'll become!

  1. Recognize the Exponential Form: Identify the equation as exponential and look for opportunities to simplify.
  2. Make a Substitution: Use a substitution (like y = 3^x) to transform the equation into a more manageable form, such as a quadratic equation.
  3. Solve the Simplified Equation: Solve the quadratic equation using factoring, the quadratic formula, or other appropriate methods.
  4. Substitute Back: Replace the substituted variable with its original expression and solve for the original variable.
  5. Verify the Solutions: Plug the solutions back into the original equation to check their validity.

Tips and Tricks for Solving Exponential Equations

Here are some extra tips and tricks to keep in mind when solving exponential equations. These can save you time and prevent common mistakes. Think of these as the secret sauce to mastering exponential equations!

  • Look for Common Bases: Try to express all exponential terms with the same base. This often simplifies the equation and makes it easier to solve.
  • Use Logarithms: If you can't easily express the terms with the same base, logarithms are your best friend. Remember the properties of logarithms, such as log_b(a^c) = c * log_b(a), which can help you bring the exponent down.
  • Be Careful with Extraneous Solutions: Always check your solutions, especially when using logarithms, as they can sometimes introduce extraneous solutions.
  • Practice, Practice, Practice: The more exponential equations you solve, the more comfortable you'll become with the techniques involved. Don't be afraid to try different approaches and learn from your mistakes.

Conclusion

So, there you have it! We successfully solved the exponential equation 3*(9^x) - 10*(3^x) + 3 = 0 by using substitution, factoring, and a bit of algebraic manipulation. Remember, the key to solving these types of problems is to break them down into smaller, more manageable steps. With practice, you'll become a pro at tackling exponential equations!

Keep practicing, and happy solving, guys! And remember, math can be fun when you approach it with the right mindset and tools. Keep exploring and keep learning!