Oh god, here comes another math post! I can feel it coming on, like werewolfism during the full moon.

I'm only passingly familiar with logarithms, so this, like everything I've stumbled on, has probably already been discovered, but

n/(1/((n^(1/n))-1))

Is a pretty good approximation (within a couple percentage points, or three or more digits) of the natural logarithm for all the numbers I've checked it on.

For example if

n = 690841693

ln(n) = 20.35342125707679

while our estimate using the above formula comes out to:

n/(1/((n**(1/n))-1)) = 20.353421612948146

Am I missing something obvious here, and if so, what?

Am I doing the idiot savant thing again, or am I just being an idiot again?