2^7=2*2*2*2*2*2*2=128 poses no practical problem.
But 3^4567 does!
From my previous question (18th Jan) and kind answer I have learned that 3^4567=10^x, and from that equation we can deduce that x=4568/log_3 10.
If we insert these values to calculator, we can find that answer is about 4568/2,095...=2180,4296... which means that there are 2180 integers in an answer if we "open" 3^4567.
But where does this denominator value log_3 10 comes from?
What kind of a number it and other log values are?
Are they always decimal numbers? ...irrational numbers? ...transcendental numbers?
As far as I know Mr. Napier (who invented logarithms) and his contemporaries had to use a lot of time to find approximate values for logarithms they used...and even so, they only tabulated some 5 decimals for each case.
So, what kind is an algorithm that creates logaritms nowadays?