The inequality is  $1000n^3&lt;2^n$ I'm trying to see for what value of $n$ an algorithm A who takes ($1000n^3$) steps takes less time than another algorithm B who takes($2^n$) steps.  Any idea how to do that step by step?
I'm new to this website. I've checked the other topics and couldn't find the answer to my question. I've also tried to find the result with TI-CAS but the only thing it does is simplify the inequality which really doesn't help at all.  the equation is : $1000n^3 &lt; 2^n$ Now I've been trying to use log but it gets me nowhere (I haven't seen this form before), the answer is supposed to be 24 but I'm not sure. So my question is how to solve step by step this inequality for n. Could you guys help me with this one?
When you use the allowframebreaks option in beamer then you get additional frames if the content is too high for one frame. The vertical spacing gets disturbed when you use a list (itemize, enumerate, description), for example, see the following animation. I used the t option (for top alignment) for the frame.     \documentclass[aspectratio=169]{beamer}   \begin{document}  \begin{frame}[t,allowframebreaks] \frametitle{Frame Title} \setlength{\mylength}{\baselineskip} \begin{itemize} \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \end{itemize}  \end{frame}  \begin{frame}[t,allowframebreaks] \frametitle{Frame Title}  Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\ Text\\  \end{frame}  \end{document}    I thought of a solution, when I read  question where they explain the different length in lists. The main idea is to check if the current frame is not the first frame of a frame continuation and add the needed vertical space. I thought that \topsep is the needed length but apparently, it is not quite right. It ist almost perfect but it still "dances" a little bit. Which additional length do I need? Here's the animation and the used code.     \documentclass[aspectratio=169]{beamer}  \setbeamertemplate{frametitle continuation}{%     (\insertcontinuationcount)%     \ifnum\insertcontinuationcount&gt;1%     \vspace*{\topsep}%     \else%     %     \fi% }  \begin{document}  \begin{frame}[t,allowframebreaks] \frametitle{Frame Title}  \begin{itemize} \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \item Test \end{itemize}  \end{frame}  \end{document}   Related        Definition of \itemize in beamer  In beamerbaselocalstructure.sty I found  \renewcommand{\itemize}[1][]{%   \beamer@ifempty{#1}{}{\def\beamer@defaultospec{#1}}%   \ifnum \@itemdepth &gt;2\relax\@toodeep\else     \advance\@itemdepth\@ne     \beamer@computepref\@itemdepth% sets \beameritemnestingprefix     \usebeamerfont{itemize/enumerate \beameritemnestingprefix body}%     \usebeamercolor[fg]{itemize/enumerate \beameritemnestingprefix body}%     \usebeamertemplate{itemize/enumerate \beameritemnestingprefix body begin}%     \list       {\usebeamertemplate{itemize \beameritemnestingprefix item}}       {\def\makelabel##1{%           {%             \hss\llap{{%                 \usebeamerfont*{itemize \beameritemnestingprefix item}%                 \usebeamercolor[fg]{itemize \beameritemnestingprefix item}##1}}%           }%         }%       }   \fi%   \beamer@cramped%   \raggedright%   \beamer@firstlineitemizeunskip% }   But I do not understand where the different kinds of spaces are defined that cause the "dancing".  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