Bits Please!

I counted to number of digits, since we can represent the number of powers of 10. There were 39 digits, 38 discounting a power to the zero. So next, we can use the change of base formula: $\frac {38}{log (2)}\approx 126.23$ . All that is needed now it to find a power of 2 that is closest to 126. 128 is $2^7$ , so we get the answer.

For a number to be represented in binary representation, we need $log_2 (n)$ bits. And hence to represent such a number 'n' in the problem, We actually need 128 bits. Yes, Vijay is in need of 128 bit computer systems for his software company.

If there are 'n' bits,then the number 'x' that can be represented using these 'n' bits can be,
$x = 2^n$

This problem asks for n, if x is 340282366920938463463374607431768211456,

Then $log_2 (x) = log_2 (2^n)$
$log_2 (x) = n$

And hence we need $\boxed{log_2 (x)}$ bits to represent a number 'x'.

Thanks for Solving...!!!