 # Alternative Logarithms

Mar 21st, 2020
657
Never
Not a member of Pastebin yet? Sign Up, it unlocks many cool features!
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.
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 * s(a)
29.
30. s(a^3) = sqrt[ s(a^2)^2 + s(a)^2 ] = sqrt[ 3 * s(a)^2 ] = sqrt * s(a)
31.
32. s(a^4) = sqrt[ 2 * s(a)^2 + 2 * s(a)^2 ] = sqrt * 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
RAW Paste Data