%5Ex%2B2%3D256%5Ex.jpeg "(1/4)^x+2=256^x")
Solving the Exponential Equation: (1/4)^x+2 = 256^x
This article will explore the solution to the exponential equation (1/4)^x+2 = 256^x. We will utilize the properties of exponents to simplify the equation and ultimately solve for the value of 'x'.
Understanding the Equation
The equation involves two terms with exponents. To simplify it, we need to express both bases (1/4 and 256) as powers of the same number.
Notice that both 1/4 and 256 can be expressed as powers of 4:
- 1/4 = 4^(-1)
- 256 = 4^4
Rewriting the Equation
Let's substitute these values into the original equation:
(4^(-1))^x+2 = (4^4)^x
Using the power of a power rule [(a^m)^n = a^(m*n)], we simplify further:
4^(-x-2) = 4^(4x)
Solving for x
Now that the bases are the same, we can equate the exponents:
-x - 2 = 4x
Combining like terms:
5x = -2
Finally, solving for x:
x = -2/5
Conclusion
Therefore, the solution to the exponential equation (1/4)^x+2 = 256^x is x = -2/5. By understanding the properties of exponents and applying them strategically, we were able to simplify the equation and isolate the variable 'x'.