Problem in determining logarithmic singularities

[Yathartha Joshi]

Consider the following logarithmic equation:

log(3*x) - log(1  - x) - log(4*x + 1)

Wolfram alpha gives the result as `-1/2` and `1/2`, but mentions `-1/2` is only valid when considering complex valued logarithm.

So to solve this equation in the `Real` domain, should `-1/2` be considered as a solution because substituting it in the equation will give log with negative argument and I suppose (I am not sure though) it should not be considered in the real domain

When these solutions are checked using checksol() the result is `True` for both the cases because the `I*pi` terms get canceled out.

A brief background of this situation:

I was implementing log solver for `_transolve` in #14792 and the idea is to convert the equation into a single instance of `log` (using `logcombine`) which can further be solved by passing it to `_solveset`, but result returned from `solveset` can have unwanted solutions (see this test). To remove such solutions `checksol()` can be used but it possesses a problem as mentioned above, so I tried a different approach to remove these solutions (see here)

The problem is what should be done here, should such solutions be considered? If not is the approach used efficient enough to remove logarithmic singularities?

[Aaron Meurer]

I would consider -1/2 to be valid in the real domain. -1/2 is a real
number and plugging it into the equation produces 0, so it's a
solution.

I think the domain keyword should only be concerned with filtering
solutions. If we also use it to change the meaning of functions in the
expression things start getting more difficult.

You can get the same problem with square roots. Consider
solveset(sqrt(x + 1)/sqrt(x - 1), domain=Reals). The solution ({-1})
when plugged in, gives 0/(2*I), which involves complex numbers. So
should solveset(domain=Reals) change the definition of sqrt to only be
valid for positive numbers?

One of the reasons we default to complex numbers in SymPy is that it
simplifies things everywhere. We don't have to worry about an
algorithm doing something that isn't valid for a restricted "reals
only" function because it somehow goes through the complex plane.

Aaron Meurer

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[Yathartha Joshi]

> I think the domain keyword should only be concerned with filtering solutions.

So it means that `solveset` should not check the solutions by plugging into the function, just check the domain of the solutions. Right?

 What should be the solution set for 

log(x - 3) + log(x + 3)

We have `sqrt(10), -sqrt(10)` as the real solutions. And as per the above statement, `solveset` should return both of these solutions.

But `solve` as well as WolframAlpha. considers only  `sqrt(10)`.

> So should solveset(domain=Reals) change the definition of sqrt to only be valid for positive numbers?

I am not sure but interestingly WolframAplha says no solutions exists for the equation `sqrt(x + 1)/sqrt(x - 1)`.