Separating variables *a* and *b* in a transcendental equation

Questions
,
1

eq1 = 2 (a + b) == E^(2 a) + 2 Log[b] + 1

How to transform eq1 into the form of eq2:

eq2 = E^(2 a) - 2 a + 1 == 2 b - 2 Log[b]

The transformation rule is to move terms containing the same variable to the same side of the equation. Specifically for this problem, terms with the letter ‘a’ should be moved to one side, while terms with ‘b’ should be moved to the other side.

2

You can use AddSides or SubtractSides.

3

Assuming Expand[] will separate terms as it does in eq1:

eq1 = 2 (a + b) == E^(2 a) + 2 Log[b] + 1;
GatherBy[List @@ Expand[Subtract @@ eq1], FreeQ[b]] //
 Replace[{{lhs_, rhs_} :> Plus @@ lhs == -Plus @@ rhs,
   {oneside_} :> Plus @@ oneside == 0}]

(*  -1 + 2 a - E^(2 a) == -2 b + 2 Log[b]  *)
4

Here is a way using SubtractSides:

eq1 = 2 (a + b) == E^(2 a) + 2 Log[b] + 1 // Expand
SubtractSides[eq1,
 Select[eq1[[1]], FreeQ[a]] +
  Select[eq1[[2]], FreeQ[b]]]
5

Okay, these two functions are useful for transposing the equation.

6

Perfect！ thanks to your approach—the issue is now resolved.

7

Thanks for your assistance; the issue is now solved.