How does $n < 2^n$ become $\log n < n$ by taking log of both sides?

Question

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...

Answer 1

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 $\log_2(n) < n\log_2(2) = n$

Answer 2

$ n < 2^n $ then $\log n <n \log 2$, since $\log 2 < 1$, we claim that $$\log n < n$$