# How to prove ln(n) = Θ(log2 n)?

This is a homework problem and I'm not sure how to do it correctly. It says "Prove ln(n) = Θ(log2 n) with n = odd number".

Bu using Natural logarithm rules, we can somehow know this is partcially true

However, I'm stuck with this since I don't understand how to do the whole procedure.

• Hint: $\frac{1}{\log_2 e}$ is a constant. – Pseudonym Sep 1 '20 at 5:54
• Take the definition of "Big Theta", choose suitable constants, prove. (If something is "obvious", by contradiction (to the negation) may work.) – greybeard Sep 1 '20 at 20:56

## 1 Answer

You correctly pointed out that $$\ln n = \frac{\log_2n}{\log_2 e}$$. You can rewrite this as $$\ln n = \frac{1}{\log_2e}\cdot\log_2n$$. Since $$\Theta(\cdot)$$ allows you to drop constant factors (and as @Pseudonym pointed out, $$\frac{1}{\log_2e}$$ is constant), it follows that $$\ln n = \Theta(\log_2n)$$