Solving the Exponential Equation (1/4)^x = 64
This article will guide you through the process of solving the exponential equation (1/4)^x = 64. We will utilize the properties of exponents and logarithms to find the solution for x.
Understanding the Problem
The equation (1/4)^x = 64 represents an exponential relationship where the base is 1/4 and the exponent is x. We need to determine the value of x that satisfies this equation.
Solving the Equation

Express both sides with the same base:
 Notice that 64 can be written as 4^3 (4 * 4 * 4 = 64).
 Also, 1/4 can be written as 4^1 (a negative exponent indicates the reciprocal of the base).
 Therefore, we can rewrite the equation as: (4^1)^x = 4^3.

Simplify using exponent rules:
 When raising a power to another power, we multiply the exponents: (4)^(x) = 4^3.

Equate the exponents:
 Now that both sides have the same base, we can equate the exponents: x = 3.

Solve for x:
 Multiply both sides by 1 to isolate x: x = 3.
Conclusion
Therefore, the solution to the equation (1/4)^x = 64 is x = 3. We have successfully solved the exponential equation by expressing both sides with the same base and applying the properties of exponents.