Formal substitution of exponential factors in an expression

Question

Assume that we have some complex algebraic expression, like

Exp[(a t + b s)/w] ( t Exp[ q t] w + q Exp[q s] Exp[-2 s] + a^2 Exp[3 s] + s t + 1 )

Now, this is just an example, the form of the expression is not very important. What is important is that we want to understand which is the exponential dependence on

Exp[s] and Exp[t]

by using some symbols, say es and et, as placeholders for Exp[s] and Exp[t].

In practice, I'd like to rearrange it to obtain (formally):

et^(a/w) es^(b/w) ( t et^q w + q es^(q-2) + a^2 es^3 + s t + 1 )

Note that I can not use the substitution /.{t -> Log[et]} because t and s also appear outside the exponential functions.

Answer 1

$Version

(* "12.3.1 for Mac OS X x86 (64-bit) (June 19, 2021)" *)

Clear["Global`*"]

expr = Exp[(a t + b s)/w] (t Exp[q t] w + q Exp[q s] Exp[-2 s] + 
     a^2 Exp[3 s] + s t + 1);

expr //. {Exp[c_. (a_. s + b_.)] :> es^(a*c)*Exp[b*c], 
  Exp[c_. (a_. t + b_.)] :> et^(a*c)*Exp[b*c]}

(* es^(b/w) et^(a/w) (1 + a^2 es^3 + es^(-2 + q) q + s t + et^q t w) *)

Note that the first factor is es^(b/w) NOT (as you showed) es^(a/w)

Answer 2

Another workable solution that uses your idea of using Log:

expr /. Exp[a_] :> Simplify@Exp[Expand@a /. {t -> Log[et], s -> Log[es]}]

Wrap it up into a function where you specify the variables:

f[expr_, vars_List] := expr /. Exp[a_] :> Simplify@Exp[Expand@a /. Thread[vars -> Log[Symbol["e" <> ToString[#]] & /@ vars]]]

Then:

expr = Exp[(a t + b s)/w] ( t Exp[ q t] w + q Exp[q s] Exp[-2 s] + a^2 Exp[3 s] + s t + 1 );
f[expr, {s, t}]
(* es^(b/w) et^(a/w) (1 + a^2 es^3 + es^(-2 + q) q + s t + et^q t w) *)

Although, I prefer to index variables, so I would use

f[expr_, vars_List] := expr /. Exp[a_] :> Simplify@Exp[Expand@a /. Thread[vars -> Log[e /@ vars]]]

instead.