Hi,

I hope someone can help.

I'm trying to figure out how to formally prove that $$\log_2(n) - \log_{10}(n) \geq \frac{\log_2(n)}{2^n}$$

I know intuitively that the right-hand side will become 0 for very large values of n, and the left-hand side will be increasing, but I am not sure how to prove this. Any ideas?

Below I have the following definitions which may or not be helpful in proving:

increasing function: $$\forall x_1, x_2 \in D, x_1 < x_2 \Rightarrow h(x_1) < h(x_2)$$

delta-epsilon definition as x approaches infinity: $$\forall \epsilon > 0, \exists M \in \mathbb{R}, x > M \Rightarrow |f(x) - L| < \epsilon$$

(Also, sorry if the title of this post was not that descriptive. I did not once mention an $n_0$ in my question here so I apologize!)