Exponential expression is not factored.

[proy87]

factor(4**x-2**x-2) does nothing.

[oscarbenjamin]

The problem is that Poly treats both 2**x and 4**x is algebraically independent generators:

In [16]: Poly(4**x - 2**x - 2)                                                                                         
Out[16]: Poly(-(2**x) + (4**x) - 2, 2**x, 4**x, domain='ZZ')

Even if told the generator it can not spot the relationship between 2**x and 4**x:

In [24]: Poly(4**x - 2**x - 2, 2**x)                                                                                   
Out[24]: Poly(-(2**x) + 4**x - 2, 2**x, domain='ZZ[4**x]')

If 4**x is instead written as (2**x)**2 then it works:

In [23]: factor((2**x)**2 - 2**x - 2)                                                                                  
Out[23]: 
⎛ x    ⎞ ⎛ x    ⎞
⎝2  - 2⎠⋅⎝2  + 1⎠

I think the implementation of Poly in identifying generators can be improved to detect algebraic relationships among the generators.

[sylee957]

I think that powdenest that is used by exponential equation solvers in sympy, is able to decompose the exponential equation into convenient forms

>>> powdenest(8**x + (S(1)/4)**(x+1) - 12)
2**(3*x) + 2**(-2*x - 2) - 12
>>> powdenest(8**x + sqrt(2)*2**(x+1) - 12)
2**(3*x) + 2**(x + 3/2) - 12

But I don't think that it makes stuff that much canonical if rational or polynomial terms are involved in the base e.g. (S(3)/2)**x + (2/S(3))**x - 12 and a**x + (a**2)**x - 12

And I'm even finding it annoying that there are relatively poor studies done on the topic of solving exponential equations in more computationally stable way via substitutions with polynomial terms, besides the materials that are taught for precalculus level students.

[smichr]

solving exponential equations in more computationally stable way via substitutions with polynomial terms

Could you please give an example of this?

[smichr]

using #22267

>>> generators(4**x-2**x-2)
{2**x}
>>> (4**x-2**x-2).subs(_.pop(),y)
y**2 - y - 2
>>> factor(_)
(y - 2)*(y + 1)

and also

>>> generators(8**x + (S(1)/4)**(x+1) - 12)
{2**(-x - 1), 2**x}  <--------- should that be {2**-x, 2**x}?
>>> generators((S(3)/2)**x + (2/S(3))**x - 12)
{(3/2)**x}
>>> generators(a**x + (a**2)**x - 12)
{a**x, (a**2)**x}
>>> a = Symbol('a', nonnegative=True)
>>> generators(a**x + (a**2)**x - 12)
{a**x}