(25)^x-1=64(5/2)^6

Jasonbradley

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Jun 16, 2024

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Solving the Exponential Equation: (25)^x-1 = 64(5/2)^6

This article will guide you through solving the exponential equation (25)^x-1 = 64(5/2)^6. We will utilize the properties of exponents and logarithms to arrive at the solution.

Simplifying the Equation

1. Express both sides with the same base:
   • We can rewrite 25 as 5^2 and 64 as 2^6.
   • The equation now becomes: (5^2)^(x-1) = 2^6 (5/2)^6.
2. Apply exponent rules:
   • (a^m)^n = a^(m*n)
   • The equation becomes: 5^(2x-2) = 2^6 * 5^6 / 2^6
3. Simplify further:
   • The 2^6 terms cancel out, leaving: 5^(2x-2) = 5^6

Solving for x

1. Equate the exponents:
   • Since the bases are the same, we can equate the exponents: 2x - 2 = 6.
2. Solve the linear equation:
   • Add 2 to both sides: 2x = 8.
   • Divide both sides by 2: x = 4.

Conclusion

Therefore, the solution to the exponential equation (25)^x-1 = 64(5/2)^6 is x = 4.