Two aligning indexes in equation

Question

I would like to align the \Longleftrightarrow's with each other, such that the second one is in the same vertical space as the first one, and to not tamper with the aligning of the equal signs (the equals sign in the line with the second \Longleftrightarrow isn't in question).

I'm not sure how to use the ampersands to align both, but in two different places.

\begin{split}
    a^a = b^b \Longleftrightarrow a & = \log_a{b^b} \\
    & = b\log_a{b}. \\
    & = \frac{b\ln b}{\ln a} \\
    \Longleftrightarrow a\ln a = b\ln b.
\end{split}

Answer 1

You can align your split to the arrows, and use the aligned environment to align three lines by =. An optional argument helps with correct positioning of the aligned rows:

\documentclass{article}
\usepackage{mathtools}
\begin{document}
\[
\begin{split}
  a^a = b^b &\Longleftrightarrow
  \begin{aligned}[t]
  a & = \log_a{b^b} \\
    & = b\log_a{b}. \\
    & = \frac{b\ln b}{\ln a}
  \end{aligned}\\
    &\Longleftrightarrow a\ln a = b\ln b.
\end{split}
\]
\end{document}