Correctness of asymptotic expression from exact expression

Question

I have the expression: \begin{equation} |Q|f(n)|\Gamma|^{f(n)} \end{equation}

Here is my solution to convert the above into an asymptotic expression: $|Q| = 2^l$ for some $l\in\mathbb{R}$ $|\Gamma| = 2^k$ for some $k\in\mathbb{R}$

Therefore we have $2^lf(n)(2^k)^{f(n)} = f(n)2^{kf(n)+l} = f(n)2^{O(f(n))}$

1. Is this correct? I just want to verify my understanding of a discussion in Sipser's Theory of Computation text.

Answer 1

What you wrote is correct under the assumption that $|\Gamma|$ and $|Q|$ are constant (which, from the context, is indeed the case), but can be taken a bit further, and can be made more explicit:

For every $x>0$ it holds that $x=2^{\log x}$. Therefore you get

$$f(n)|Q||\Gamma|^{f(n)}=2^{\log f(n)}2^{\log |Q|}2^{f(n)\log |\Gamma|}= 2^{\log f(n)+\log |Q|+f(n)\log |\Gamma|}=2^{O(f(n))}$$

which is somewhat cleaner.