Why is $3^n = 2^{O(n)}$ true?

Question

$3^n = 2^{O(n)}$ is apparently true. I thought that it was false though because $3^n$ grows faster than any exponential function with a base of 2.

How is $3^n = 2^{O(n)}$ true?

Answer 1

With some algebra (and changing the constant in the $O(n)$), we can actually change the bases.

$$3^n = (2^{\log_2 3})^n = 2^{n\log_2 3}$$

Since $\log_2 3$ is a constant, $n\log_2 3 = O(n)$. So $3^n = 2^{O(n)}$.

I'm not sure what you mean by "$3^n$ grows faster than any exponential function with a base of 2." $2^n = o(3^n)$ of course but it seems you mean something more general. My guess is that your statement applies to something like $O(3^n)$, where you multiply the base by a constant, as opposed to $2^{O(n)}$ where you multiply the number in the exponent by a constant.

Answer 2

$3^n$ grows faster than any exponential function with a base of $2$.

True. This implies that $3^n = O(2^n)$ cannot be true. But what you have here is $2^{O(n)}$.

Recall that $O(f(n))$ is really a set of functions, and strictly speaking we should be writing $3^n \in 2^{O(n)}$ (or even $(n \mapsto 3^n) \in 2^{O(n \mapsto n)}$). The right-hand side is not the exponential of a function, but the exponential of a set of functions. Expanding the definition of big oh:

$$ 2^{O(n)} = 2^{\{f \;\mid\; \exists N, \exists p, \forall n \ge N, f(n) \le p n\}} = \{(n \mapsto 2^{f(n)}) \mid \exists N, \exists p, \forall n \ge N, f(n) \le p n\} $$

Since the exponential function $n \mapsto 2^n$ is increasing, we can lift the inequality out of the exponential:

$$ 2^{O(n)} = \{g \mid \exists N, \exists p, \forall n \ge N, g(n) \le 2^{p \, n}\} $$

Contrast with $$ O(2^n) = \{g \mid \exists N, \exists k, \forall n \ge N, g(n) \le k \, 2^{n}\} $$

In $2^{O(n)}$, the multiplicative constant is inside the exponential. In $O(2^n)$, it is multiplied by the exponential. $2^{p \, n} = 2^p 2^n$, so we have (for any $n \ge 0$) $3^n \le 2^{\log_2 3} 2^n$, i.e. we can take $N = 0$ and $p = \log_2 3$, showing that $3^n \in 2^{O(n)}$.

Answer 3

$3^n=2^{O(n)}$ is in fact true because if you recall $O(n)$ definition you will see that you can add/multiply by any constant. So:

$3^n < 2^{kn}$ // $ \log_2 $

$n\log_2(3^n) < kn\log_2(2)$

$k > \log_2(3) $

So as you can see $2^{kn}$ is bigger then $3^n \iff k > \log_2(3)$