Why does f ( x y ) = f ( x ) + f ( y ) ⟹ f ( x ) = k ln ⁡ xMy textbook says f ( x y ) = f ( x ) + f ( y ) ⟹ f ( x ) = k ln ⁡ xBut if I'm not mistaken, f(x) could be a logarithmic function with any base? So why only ln ⁡ x?

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Why does   f  (  x  y  )  =  f  (  x  )  +  f  (  y  )    ⟹    f  (  x  )  =  k  ln  ⁡  xMy textbook says   f  (  x  y  )  =  f  (  x  )  +  f  (  y  )    ⟹    f  (  x  )  =  k  ln  ⁡  xBut if I&#039;m not mistaken, f(x) could be a logarithmic function with any base? So why only   ln  ⁡  x?

hocelwsmjc · Accepted Answer

This is false. If we let   g  (  x  )  =  f  (      e    x    )  , then we find that g satisfies Cauchy&#039;s functional equation, and hence there are infinitely many pathological g which correspond to solutions   f  =  g  (  ln  ⁡  x  )  . You must place some continuity or differentiability restrictions on f.

Layton Park · Accepted Answer

Because       log    a    ⁡  x  =      1          ln      ⁡      a        ln  ⁡  x. So, a generic logarithmic function is always a multiple of the natural logarithm.

Why does f(xy)=f(x)+f(y) =>f(x)=k lnx

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