Alternative Logarithms

Alternative Logarithms, Part 1
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We are familiar with the logarithm identity

  Log(ab) = Log(a) + Log(b)

What happens if we adjust this rule slightly?

  s(ab) = f( s(a), s(b) ) for some symmetric function f.

eg, Instead of adding, consider the L2 metric:

  s(ab) = sqrt[ s(a)^2 + s(b)^2 ]

Can we determine a closed form of s() from this rule?

Let us evaluate it at some points:

  s(a.1) = sqrt[ s(a)^2 + s(1)^2 ] = s(a)

Clealy s(1)^2 must be zero, ie, s(1) = 0

  s(a.0) = sqrt[ s(a)^2 + s(0)^2 ] = s(0)

Clearly s(x) --> inf as x --> 0, in order to make the s(a)^2 term insignificant, ie, s(0) = inf

  s(a^2) = sqrt[ 2 * s(a)^2 ] = sqrt[2] * s(a)

  s(a^3) = sqrt[ s(a^2)^2 + s(a)^2 ] = sqrt[ 3 * s(a)^2 ] = sqrt[3] * s(a)

  s(a^4) = sqrt[ 2 * s(a)^2 + 2 * s(a)^2 ] = sqrt[4] * s(a)

And in general, s(a^n) = sqrt[n] * s(a)

We can use this expression to obtain a parameterisation of s. Let a be constant, and allow n to vary.

Then points (x,y) on the graph of s() satisfy the parameterisation:

  x = a^n, and y = sqrt[n] * s(a)    // see that s(a) is also a constant

  Therefore n = log(x)/log(a), and so y = sqrt[log(x)] * constant

For sake of simplicity we can choose 'a' such that this constant = 1

  s(x) = sqrt[log(x)]
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Verifying:

  s(x)^2 = log(x)

  s(ab) = sqrt[log(ab)] = sqrt[log(a) + log(b)] = sqrt[s(a)^2 + s(b)^2]

We can consider a family of such alternative logarithms, with parameter r:

  s(ab;r) = [ s(a)^r + s(b)^r ] ^ (1/r)

which by similar arguments are satisfied by:

  s(x;r) = log(x)^(1/r)

Incidentally, this right hand expression [ A^p + B^p ] ^ (1/p) is the Lp-norm, for p>=1, and A,B>=0