Show that there does not exist a strictly increasing function f:N→N satisfying f(2)=3 and f(mn)=f(m)f(n) for all m,n∈N.
My Attempt
It can be found that f(2x)=3x from which we can conclude that f(x)=3log2x and by which we can argue that for all natural x, f(x) is not always natural, so contradiction hence such a function doesn't exist.
Is this proof correct? If not what should be changed?
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