Lots of ideas however mostly not working, for example:

Assume there would be a function with

Then surely

So if there is an with we have

.

for we also have:

and both together we get

If we assume that is continuous then this is valid not only for rationals but also for reals :

.

No let constant and let be variable:

So for some constant . But then:

if :

which can not be satisfied for all , .

Hence either or which we excluded in our previous considerations.

Proposition. Let be a continuous function defined on the positive reals such that for all , then either for all or for all .

Ivars Wrote:3_log(a^b) =3_log(a[3]b)= 3_log(a)*3_log(b) = 3_log(a)[2]3_log(b)

Assume there would be a function with

Then surely

So if there is an with we have

.

for we also have:

and both together we get

If we assume that is continuous then this is valid not only for rationals but also for reals :

.

No let constant and let be variable:

So for some constant . But then:

if :

which can not be satisfied for all , .

Hence either or which we excluded in our previous considerations.

Proposition. Let be a continuous function defined on the positive reals such that for all , then either for all or for all .