(1/4)^x=64

Jasonbradley

2 min read

·

Jun 16, 2024

10

Share

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

1. 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.

2. Simplify using exponent rules:

   • When raising a power to another power, we multiply the exponents: (4)^(-x) = 4^3.

3. Equate the exponents:

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

4. 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.