Finding running time Apologies for this simple question. I found it in the book Algorithms by Sedgewick and Wayne.

Give a formula to predict the running time of a program for a problem of size N when doubling experiments have shown that the doubling factor is $2^b$ and the running time for problems of size N0 is T.

$$ \frac{T(2N)}{T(N)}=2^b.$$ 

$$ T(N_0)=T. $$

I am not sure how to proceed after that.
Instead of solving the question for you, let me show you what steps you need to take in order to solve this problem on your own.

Suppose that $T(1) = T_0$ and $T(2n) = C T(n)$ for all $n$.


What is $T(2)$? What is $T(4)$? What is $T(8)$? What is $T(2^k)$?


Now suppose that $T$ is monotone: $T(n_1) \leq T(n_2)$ whenever $n_1 \leq n_2$.


If $2^k \leq n \leq 2^{k+1}$, what can you say about $T(n)$?
Can you now deduce the asymptotic rate of growth of $T(n)$ for arbitrary $n$?