(1/2)^x=16

2 min read Jun 16, 2024

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

This article explores the process of solving the exponential equation (1/2)^x = 16.

Understanding the Equation

The equation (1/2)^x = 16 presents a challenge: we need to find the value of 'x' that makes the equation true. To solve this, we'll utilize the properties of exponents and logarithms.

Steps to Solve the Equation

Express 16 as a power of 1/2: Since 16 is 2^4, we can rewrite the equation as: (1/2)^x = (1/2)^(-4)
Equate the exponents: When the bases of the exponents are the same, we can directly equate the exponents. This gives us: x = -4

Solution

Therefore, the solution to the equation (1/2)^x = 16 is x = -4.

Verification

We can verify our solution by substituting x = -4 back into the original equation:

(1/2)^(-4) = 1/(1/2)^4 = 1/(1/16) = 16

This confirms that our solution, x = -4, is correct.

Conclusion

Solving exponential equations requires understanding the properties of exponents and logarithms. By applying these principles, we can effectively solve equations like (1/2)^x = 16 and determine the value of the unknown variable.