1
$\begingroup$

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

enter image description here

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

$\endgroup$
2
  • 1
    $\begingroup$ Hint: $\frac{1}{\log_2 e}$ is a constant. $\endgroup$
    – Pseudonym
    Sep 1, 2020 at 5:54
  • $\begingroup$ Take the definition of "Big Theta", choose suitable constants, prove. (If something is "obvious", by contradiction (to the negation) may work.) $\endgroup$
    – greybeard
    Sep 1, 2020 at 20:56

1 Answer 1

Reset to default
1
$\begingroup$

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 $\ln n = \Theta(\log_2n)$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.