# Small-o and Theta questions

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• Sep 29th 2011, 02:03 AM
Evangeline
Small-o and Theta questions
I have these questions from a book:

show that if f(n)=theta(log2 n) then f(n)=theta (log10 n)

show that if f(n)=o(g(n)) and g(n)=o(f(n)) then f(n)=theta(g(n))

how to prove? can anyone help me with it?
• Sep 29th 2011, 02:06 AM
Siron
Re: please help me with these quetions
The first proposition, do you mean:
$\displaystyle f(n)=\theta(\log_{2}n) \Rightarrow f(n)=\theta(\log10n)$
?
• Sep 29th 2011, 01:09 PM
emakarov
Re: please help me with these quetions
To show the first statement, write the definition of $\displaystyle \Theta$ and note that $\displaystyle \log_2n=\log_210\log_{10}n$.

The second statement is a little odd because I believe that if f(n) = o(g(n)) and g(n) = o(f(n)), then f and g must be 0 from some point. However, the fact that g(n) = o(f(n)) implies that g(n) = O(f(n)), and this is almost the same as $\displaystyle f(n)=\Theta(g(n))$ (up to absolute value).
• Sep 29th 2011, 01:17 PM
Evangeline
Re: Small-o and Theta questions
Quote:

The first proposition, do you mean:
http://latex.codecogs.com/png.latex?...28%5Clog10n%29
?
yes, that's what i meant

Quote:

To show the first statement, write the definition of http://latex.codecogs.com/png.latex?%5CTheta and note that http://latex.codecogs.com/png.latex?...Clog_%7B10%7Dn.

The second statement is a little odd because I believe that if f(n) = o(g(n)) and g(n) = o(f(n)), then f and g must be 0 from some point. However, the fact that g(n) = o(f(n)) implies that g(n) = O(f(n)), and this is almost the same as http://latex.codecogs.com/png.latex?...%28g%28n%29%29 (up to absolute value).
thanks for help :)