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f(n) = log_p n if f(kl) = f(k)+ f(l)

Source: Polish MO Finals 1976 p6

August 23, 2024
algebralogarithmfunctional

Problem Statement

An increasing function f:N→Rf : N \to R satisfies f(kl)=f(k)+f(l)   for   all   k,l∈N.f(kl) = f(k)+ f(l)\,\,\, for \,\,\, all \,\,\, k,l \in N. Show that there is a real number p>1p > 1 such that f(n)= logpnf(n) =\ log_pn for all nn.
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