(1/4)^3z-1=16^z+2*64^z-2

Jasonbradley

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Jun 16, 2024

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Solving the Exponential Equation: (1/4)^(3z-1) = 16^(z+2) * 64^(z-2)

This article explores the process of solving the given exponential equation. We'll utilize the properties of exponents and logarithms to isolate the variable z and arrive at a solution.

Understanding the Problem

The equation presented involves multiple terms with different bases raised to various exponents. Our goal is to simplify the equation by expressing all terms with the same base, enabling us to solve for z.

Solution

1. Express all bases as powers of the same number:

   • We notice that all bases (1/4, 16, and 64) can be expressed as powers of 4:
     • (1/4) = 4^(-1)
     • 16 = 4^2
     • 64 = 4^3

2. Substitute the equivalent bases into the equation:

   • (4^(-1))^(3z-1) = (4^2)^(z+2) * (4^3)^(z-2)

3. Apply the power of a power rule:

   • 4^(-3z+1) = 4^(2z+4) * 4^(3z-6)

4. Combine terms with the same base:

   • 4^(-3z+1) = 4^(2z+4 + 3z-6)
   • 4^(-3z+1) = 4^(5z-2)

5. Equate the exponents:

   • -3z + 1 = 5z - 2

6. Solve for z:

   • 3 = 8z
   • z = 3/8

Verification

To ensure our solution is correct, we can substitute z = 3/8 back into the original equation and check if both sides are equal.

Conclusion

We have successfully solved the exponential equation (1/4)^(3z-1) = 16^(z+2) * 64^(z-2) by utilizing the properties of exponents and simplifying the equation to a linear form. The solution is z = 3/8.