Log2(5) is 2^x = 5.
The integer part would be 2 because 2^2 = 4. And what's left is 2^(2+x) = 5. So 2^x = 5/4
We know that x actually has to be between 2 and 3 without being 3 itself because 2^3 is 8.
In order to further extract the remaining integer part (2^x = 5/4, because), instead of 2^x = 5/4 (because the whole part = 0), we can first use (5 /4)^x = 2 (which has the integer part 3) and form the reciprocal.
So then it would be Log2(5) = 2 + 1/log5/4(2)
So 2 + 1/3 is a better approximation for log2(5).
We want to get to 2, but (5/4) ³ is only 125/64. and what we now need is 2/(125/64) which is 128/125.
So (5/4)^(3+x) = 128/125
and here again because we have already taken out the whole number part (namely 3) we know what we do is we turn it around again and see how often 128/125 fits into 5/4 instead to get a better feeling. which is an integer 9.
So now we have a total
2+1/(3+1/9)
And it just goes on like that.
But the Thing is, how can i work my way Back, only knowing the Base and the logarithm value.
So to speak 2^(2+1/(3+1/9)) = x
Basically to reverse the continued fraction in a smart way ??
Or so to speak, pull the nth root only with manipulating a logarithms continued fraction