Lovely Logarithms!

Here's a function:

y = logx(2x)

If x is equal to a power of ½, then y = (n - 1)/n! (n is the exponent that you raise x by)

Note: n ≠ 0 because division by zero(0) is undefined.

Brain saying "WOW!"
xy
1/20
1/41/2
1/82/3
1/163/4
1/324/5
1/645/6
1/1286/7
1/2567/8
1/5128/9
1/10249/10
½^n(n - 1)/n

Since ½ is the reciprocal of 2, the negative powers of ½ are the positive powers of 2 & vice versa!

If x is equal to a power of 2, then y = (n + 1)/n!

Note: Again, n ≠ 0 because division by zero(0) is undefined.

xy
22
43/2
84/3
165/4
326/5
647/6
1288/7
2569/8
51210/9
102411/10
2^n(n + 1)/n

Also, remember this fact about fractions & mixed numbers:

You can also see this fact on the Main Page of the Math Section.

Technically, 1 is the zeroth power of 2 & also the zeroth power of ½, but if x = 1, then y is undefined because log1T is indeterminate. When the number 1 is the base of a logarithm, it's like dividing by zero(0)! Besides, 1T = 1, every time!

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© Derek Cumberbatch