Prove that $3\log n$ is $O(\exp(0.001n))$

Question

First time posting here. Hi math stack-exchange community!

I have a bonus question on my assignment and I am having trouble proving it. The main reason is that I am only limited to using the rules they provided me. Can anyone help me with this? I've been stuck at it for hours.

Here is where I got to: $$3log(n) = O(3n) \space -by \space Rule 7$$ $$O(3n) = O(e^{bn})\space -by \space Rule 5 \space and \space 6$$

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Below is the actual question. Prove that:

Here are the rules that I am allowed to use and also an example of how it should be formatted.

Answer 1

$$ 3log(n)=O(3n) − 7 $$

$$ O(3n)=O(e^{bn}) − 5, 6 $$

$b$, here, can be any positive number. Without loss of generality, choose $b = 0.001$

Now, by virtue of property 4,

$$ 3log(n)=O(3n) $$ and, $$ O(3n)=O(e^{0.0001n}) $$ implies that, $$3log(n) = O(e^{0.0001n}) $$