Sorceress

Alternative Logarithms

Mar 21st, 2020
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  1. Alternative Logarithms, Part 1
  2. ------------------------------
  3.  
  4. We are familiar with the logarithm identity
  5.  
  6. Log(ab) = Log(a) + Log(b)
  7.  
  8. What happens if we adjust this rule slightly?
  9.  
  10. s(ab) = f( s(a), s(b) ) for some symmetric function f.
  11.  
  12. eg, Instead of adding, consider the L2 metric:
  13.  
  14. s(ab) = sqrt[ s(a)^2 + s(b)^2 ]
  15.  
  16. Can we determine a closed form of s() from this rule?
  17.  
  18. Let us evaluate it at some points:
  19.  
  20. s(a.1) = sqrt[ s(a)^2 + s(1)^2 ] = s(a)
  21.  
  22. Clealy s(1)^2 must be zero, ie, s(1) = 0
  23.  
  24. s(a.0) = sqrt[ s(a)^2 + s(0)^2 ] = s(0)
  25.  
  26. Clearly s(x) --> inf as x --> 0, in order to make the s(a)^2 term insignificant, ie, s(0) = inf
  27.  
  28. s(a^2) = sqrt[ 2 * s(a)^2 ] = sqrt[2] * s(a)
  29.  
  30. s(a^3) = sqrt[ s(a^2)^2 + s(a)^2 ] = sqrt[ 3 * s(a)^2 ] = sqrt[3] * s(a)
  31.  
  32. s(a^4) = sqrt[ 2 * s(a)^2 + 2 * s(a)^2 ] = sqrt[4] * s(a)
  33.  
  34. And in general, s(a^n) = sqrt[n] * s(a)
  35.  
  36. We can use this expression to obtain a parameterisation of s. Let a be constant, and allow n to vary.
  37.  
  38. Then points (x,y) on the graph of s() satisfy the parameterisation:
  39.  
  40. x = a^n, and y = sqrt[n] * s(a) // see that s(a) is also a constant
  41.  
  42. Therefore n = log(x)/log(a), and so y = sqrt[log(x)] * constant
  43.  
  44. For sake of simplicity we can choose 'a' such that this constant = 1
  45.  
  46. s(x) = sqrt[log(x)]
  47. -----------------------
  48.  
  49. Verifying:
  50.  
  51. s(x)^2 = log(x)
  52.  
  53. s(ab) = sqrt[log(ab)] = sqrt[log(a) + log(b)] = sqrt[s(a)^2 + s(b)^2]
  54.  
  55. We can consider a family of such alternative logarithms, with parameter r:
  56.  
  57. s(ab;r) = [ s(a)^r + s(b)^r ] ^ (1/r)
  58.  
  59. which by similar arguments are satisfied by:
  60.  
  61. s(x;r) = log(x)^(1/r)
  62.  
  63. Incidentally, this right hand expression [ A^p + B^p ] ^ (1/p) is the Lp-norm, for p>=1, and A,B>=0
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