1. 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?

The first proposition, do you mean:
$f(n)=\theta(\log_{2}n) \Rightarrow f(n)=\theta(\log10n)$
?

To show the first statement, write the definition of $\Theta$ and note that $\log_2n=\log_210\log_{10}n$.

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 $f(n)=\Theta(g(n))$ (up to absolute value).

4. Re: Small-o and Theta questions

The first proposition, do you mean:
$f(n)=\theta(\log_{2}n) \Rightarrow f(n)=\theta(\log10n)$
?
yes, that's what i meant

To show the first statement, write the definition of $\Theta$ and note that $\log_2n=\log_210\log_{10}n$.

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 $f(n)=\Theta(g(n))$ (up to absolute value).
thanks for help