[Tex/LaTex] Beamer: Split Align Environment Across Multiple Slides

Question

I have the following slide:

\documentclass{beamer}

\mode<presentation> {
\usetheme{Madrid}
\usecolortheme[RGB={0,0,0}]{structure}
}

\usepackage{amsmath,amsfonts,graphicx}
\usepackage{algpseudocode}
\usepackage{tikz}

\pgfdeclarelayer{background}
\pgfsetlayers{background,main}
\begin{document}
\begin{frame}
\frametitle{Example: $\Omega$-Notation}
\uncover<+->{{\bf Solution:}}
\begin{align*}
\uncover<+->{\log(n!) &= \log(\prod_{i=1}^{n} i) = \sum_{i=1}^{n} \log i \\}
\uncover<+->{&\geq \sum_{i={\frac{n}{2} + 1}}^{n} \log i \qquad \forall n \geq 1 \\}
\uncover<+->{&\geq \sum_{i={\frac{n}{2} + 1}}^{n} \log{(\frac{n}{2})} \qquad \forall n \geq 1 \\}
\uncover<+->{&= \frac{n}{2} \log(\frac{n}{2}) = \frac{n}{2} (\log n - 1) \\}
\uncover<+->{&\geq \frac{n}{2}(\log n - \frac{1}{2}\log n) \qquad \forall n \geq 4 \\}
\uncover<+->{&= \frac{1}{4}n \log n \\}
\end{align*}
\end{frame}
\end{document}

The problem is that there are to many equations to fit nicely on the slide (some are omitted here). Now I would like to split this into two slides, but making sure that the equations remained aligned consistently across both slides. I have seen a few possible solutions, but none appear to work with uncover.

Answer 1

You can "fake" the alignment on a new page with \phantom{}. But as the align environment is centred in the middle of the slide, this has to be done for both sides of the equation. I think the easiest is to take the longest expressions for both sides and add phantoms of them to all slides, see the last lines in the frames.

\documentclass{beamer}

\mode<presentation> {
    \usetheme{Madrid}
    \usecolortheme[RGB={0,0,0}]{structure}
}

\usepackage{amsmath,amsfonts,graphicx}
\usepackage{algpseudocode}
\usepackage{tikz}

\pgfdeclarelayer{background}
\pgfsetlayers{background,main}
\begin{document}
    \begin{frame}[t]
        \frametitle{Example: $\Omega$-Notation}
        \uncover<+->{{\bf Solution:}}
        \begin{align*}
        \uncover<+->{\log(n!) &= \log(\prod_{i=1}^{n} i) = \sum_{i=1}^{n} \log i \\}
        \uncover<+->{&\geq \sum_{i={\frac{n}{2} + 1}}^{n} \log i \qquad \forall n \geq 1 \\}
        \uncover<+->{&\geq \sum_{i={\frac{n}{2} + 1}}^{n} \log{(\frac{n}{2})} \qquad \forall n \geq 1 \\}
        \uncover<+->{&= \frac{n}{2} \log(\frac{n}{2}) = \frac{n}{2} (\log n - 1) \\}
        \phantom{\log(n!)} & \phantom{\geq \frac{n}{2}(\log n - \frac{1}{2}\log n) \qquad \forall n \geq 4}
        \end{align*}
    \end{frame}

    \addtocounter{framenumber}{-1}

    \begin{frame}[t]
        \frametitle{Example: $\Omega$-Notation}
        \uncover<+->{{\bf Solution:}}
        \begin{align*}
        \uncover<+->{&\geq \frac{n}{2}(\log n - \frac{1}{2}\log n) \qquad \forall n \geq 4 \\}
        \uncover<+->{&= \frac{1}{4}n \log n \\}
        \phantom{\log(n!)} & \phantom{\geq \frac{n}{2}(\log n - \frac{1}{2}\log n) \qquad \forall n \geq 4}
        \end{align*}
    \end{frame} 
\end{document}