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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

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

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    $\begingroup$ 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

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

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