(1/2)^x=16^2

2 min read Jun 16, 2024

Solving the Exponential Equation: (1/2)^x = 16^2

This article will guide you through solving the exponential equation (1/2)^x = 16^2. We will utilize the properties of exponents and logarithms to find the solution for x.

Understanding the Equation

The equation presents a challenge due to the different bases of the exponents. To solve it, we need to express both sides of the equation with the same base.

Step 1: Expressing both sides with the same base

16 can be expressed as 2^4. Therefore, 16^2 is equivalent to (2^4)^2 = 2^8.
(1/2) can be expressed as 2^-1.

Substituting these values, our equation becomes: (2^-1)^x = 2^8

Applying the Power of a Power Rule

Step 2: Simplifying the equation

Using the power of a power rule, which states (a^m)^n = a^(m*n), we can simplify the left side of the equation: 2^(-x) = 2^8

Solving for x

Step 3: Equating the exponents

Since the bases are now the same, we can equate the exponents: -x = 8

Step 4: Isolating x

Multiplying both sides by -1, we get: x = -8

Solution

Therefore, the solution to the equation (1/2)^x = 16^2 is x = -8.

Verification

To verify our solution, we can substitute x = -8 back into the original equation:

(1/2)^(-8) = 16^2

(2^8) = 16^2

256 = 256

This confirms that x = -8 is the correct solution.