Wednesday, June 13, 2018

functional equations - If f(xy)=f(x)+f(y), show that f(.) can only be a logarithmic function.


As the question states, show that the property exhibited can only be satisfied by a logarithmic function i.e no other family of functions can satisfy the above property.


Answer



Continuity is necessary.


If F(x+y)=F(x)+F(y), for all x,y and F discontinuous (such F exist due to the Axiom of Choice, and in particular, the fact that R over Q possesses a Hamel basis) and f(x)=F(logx), then f(xy)=f(x)+f(y), and f is not logarithmic!


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