A googol is a 1 with 100 zeros after it.
A googolplex is 1 with a googol zeros after it.
Write googolplex in the form ee...ex with a stack of e's and x as a decimal between 0 and 1.
How many e's are there and what is x?
By definition, the common log of a googolplex is a googol. We want the natural log of a googolplex expressed as how many levels of e before the natural log is equal to (unlikely) or less than 1. Otherwise the number of e's would be arbitrary as at any point we could just make x, the highest exponent, big enough.
To convert a common log to a natural log one divides by the common log of e or multiplies by the natural log of 10. Since the common log of a googolplex is a googol or 10^100, its natural log is 10^100 / log(e) = 10^100 * ln(10) ~= 2.302585092994042 * 10^100, so e raised to that power is a googolplex. Now we just keep taking natural logs:
2.302585092994042 * 10^100
Each is the natural log of the number before it. With only one e, x would be 2.302585092994042 * 10^100, but that's not less than 1. We need 5 e's to bring x down to less than 1, and it is approximately 0.5272678974302746 or exactly ln(ln(ln(ln((10^100)*ln(10))))).
UBASIC can give more decimal places for x:
Edited on August 26, 2018, 1:42 pm
Posted by Charlie
on 2018-08-26 13:39:29