Thread:

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?

2. Re: please help me with these quetions

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

3. Re: please help me with these quetions

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:
$f%28n%29=%5Ctheta%28%5Clog_%7B2%7Dn%29%20%5CRightarrow%20f%28n%29=%5Ctheta%28%5Clog10n%29$
?
yes, that's what i meant

To show the first statement, write the definition of $%5CTheta$ and note that $%5Clog_2n=%5Clog_210%5Clog_%7B10%7Dn$.

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 $f%28n%29=%5CTheta%28g%28n%29%29$ (up to absolute value).
thanks for help
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9074801802635193, "perplexity_flag": "middle"}