Given a subfield F of $${\mathbb {C}}$$
C
, we study the linear disjointess of the field E generated by iterated exponentials of elements of $${\overline{F}}$$
F
¯
, and the field L generated by iterated logarithms, in the presence of Schanuel’s conjecture. We also obtain similar results replacing $$\exp $$
exp
by the modular j-function, under an appropriate version of Schanuel’s conjecture, where linear disjointness is replaced by a notion coming from the action of $$\textrm{GL}_2$$
GL
2
on $${\mathbb {C}}$$
C
. We also show that for certain choices of F we obtain unconditional versions of these statements.
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