How does $n < 2^n$ become $\log n < n$ by taking the log of both sides?
I understand the left side but I do not understand the right side of the inequality. The once was $\log 2^n$ becomes $n$ for some reason...
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The logarithm function is an increasing function, which means it is valid to take the $\log$ of each side of the equation.
$$\log (n) \lt \log(2^n) \iff \log(n) < n\log 2$$
This is because one of the laws of exponents tells us $$\log a^b = b\log a.$$
If you are using $\log_2$, then $\log_2(n) < n\log_2(2) = n$