Thread: Binary Integer Proof?

Hi all, any ideas with the following?

Let n be a positive integer:

Show that when n is written in binary notation, then the number k of its digits
satises

k - 1 <= ln(n)/ln(2) < k

where ln is the natural logarithm.

Thanks!

Originally Posted by sirellwood

Hi all, any ideas with the following?

Let n be a positive integer:

Show that when n is written in binary notation, then the number k of its digits
satises

k - 1 <= ln(n)/ln(2) < k

where ln is the natural logarithm.

Thanks!

Let k the positive integer s.t. 2^{k-1}<= n < 2^k . Now apply log basis 2 in the three sides and get what you want

Tonio

So could I use the same theory for when n is written in decimal notation, then the number L of its digits
satisfies:

L-1 <= ln(n)/ln(10) < L?