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(5^x+4)^x-3=0 2^x*25^x-4

4 min read Jun 16, 2024
(5^x+4)^x-3=0 2^x*25^x-4

Solving the Equation: (5^x + 4)^x-3 = 0 and 2^x * 25^x - 4

This article will guide you through solving the given equation: (5^x + 4)^x-3 = 0 and 2^x * 25^x - 4. We'll break down the steps and explain the concepts involved.

Understanding the Equation

The equation is a bit complex, with exponents within exponents. To solve it, we need to use our knowledge of exponents and algebraic manipulation.

Step 1: Simplify the Equation

First, we need to simplify the equation. Let's focus on the first part: (5^x + 4)^x-3 = 0.

  • We can rewrite 25 as 5^2. So, 2^x * 25^x - 4 becomes 2^x * (5^2)^x - 4, which simplifies to 2^x * 5^(2x) - 4.

  • Now we have:

    • (5^x + 4)^x-3 = 0
    • 2^x * 5^(2x) - 4 = 0

Step 2: Solve for x

Let's solve each part separately:

  • Part 1: (5^x + 4)^x-3 = 0

    • Case 1: 5^x + 4 = 0

      • Solving for x, we get 5^x = -4. There is no real solution for this case, as any positive number raised to any power will always be positive.

    • Case 2: x - 3 = 0

      • Solving for x, we get x = 3. This is a potential solution.

  • Part 2: 2^x * 5^(2x) - 4 = 0

    • Rewrite: 2^x * 5^(2x) = 4
    • Express 4 as a power: 4 = 2^2
    • Substitute: 2^x * 5^(2x) = 2^2
    • Simplify: 5^(2x) = 2^(2-x)
    • Take the logarithm of both sides (base 5): 2x = log5(2^(2-x))
    • Solve for x: This equation requires numerical methods (like graphing or numerical solvers) to find an approximate solution for x.

Step 3: Verify the Solutions

We found two potential solutions:

  • x = 3
  • x = (numerical solution from Part 2)

To verify if they are valid solutions, we need to plug them back into the original equation. If the equation holds true for both values, then they are valid solutions.

Conclusion

Solving the given equation involves simplifying it, considering different cases, and using appropriate techniques like logarithms and numerical methods. By carefully following these steps, we can find the potential solutions and verify their validity.

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