Math, asked by kavitha2002dvg, 4 months ago

find the value of X✓256÷✓x=1÷3​

Answers

Answered by mayurikajuthu
0

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.

Similar questions