find the value of X✓256÷✓x=1÷3
Answers
Other answers have responded to this question by pointing out that 44=256 and therefore x=4 is a potential solution to the equation. While this may be trivial or easy to solve by inspection alone, I would actually like to show how this solution may be found in terms of special functions and therefore see if any other solutions exist.
Firstly, we define the W-Lambert function as the inverse of the following function: (note this is not a elementary function, but this doesn't matter)
W−1(x)=xex
Then we return to the equation we are required to solve:
xx=256
xlnx=ln256
=lnx⋅elnx
⟹W(ln256)=lnx
Hence:
x=eW(ln256)
As it turns out, W(ln256) does only have 1 real value. However, it does also take (an infinite number of) complex values which should not be ignored such as:
0.155+4.745i
−0.681+10.933i
−1.135+17.213i
The real solution is of course ln4 which gives:
x=eln4=4
However, it is clear that there are an infinite number of other complex values which x may also take and that there is not only one solution to the equation.