Basically, use only one &
symbol per row. If this were my document, I'd stick with the default layout for the right-hand column of the cases
environment, which is left alignment of the material. For sure, you can't used an aligned
environment to achieve your formatting objective.
Two additional comments: (i) Use \left
and \right
only sparingly. (ii) Try to keep your notation consistent. At times, you use \frac{N}{n_i}
, while at others you write N/n_i
. In the present context, I'd use the latter, i.e., inline-fraction notation.

Note also that I changed all instances of tf_{ij}
in your code to \mathit{tf}_{\!ij}
, both to tighten up the appearance of tf
and to "snug up" the subscript term to its base. If you have a lot of instances of tf_{ij}
, you may want to create a shorthand macro, say, \newcommand\tfij{\mathit{tf}_{\!ij}}
, and then replace all instances of tf_{ij}
in your document with \tfij
.
\documentclass{article}
\usepackage{amsmath} % for 'cases' env. and '\text' macro
\begin{document}
\begin{equation}
\label{eq:13}
W_{ij} =
\begin{cases}
\mathit{tf}_{\!ij} \times \log(N/n_i)
& \text{if $AW_i = 0$}\\
AW_i \times \mathit{tf}_{\!ij} \times \log(N/n_i)
& \text{if $0<AW_i<1$}\\
(1 + \log AW_i) \times \mathit{tf}_{\!ij} \times \log(N/n_i)
& \text{if $AW_i \geq 1$}
\end{cases}
\end{equation}
\end{document}
Addendum: For the case at hand (pun intended), what you may want to think about, in terms of changing the layout, is to right-align the material in the first column of the cases
environment.

\documentclass{article}
\usepackage{amsmath} % for 'cases' env. and '\text' macro
\newcommand\tfij{\mathit{tf}_{\!ij}}
\begin{document}
\begin{equation}
\label{eq:13a}
W_{ij} =
\begin{cases}
\hfill \tfij \times \log(N/n_i)
& \text{if $AW_i = 0$}\\
\hfill AW_i \times \tfij \times \log(N/n_i)
& \text{if $0<AW_i<1$}\\
(1 + \log AW_i) \times \tfij \times \log(N/n_i)
& \text{if $AW_i \geq 1$}
\end{cases}
\end{equation}
\end{document}
Second addendum, to address the OP's follow-up comment: To achieve the desired special alignment of the material in the second column of the cases
environment, I suggest you use \phantom
statements in rows 1 and 3.

\documentclass{article}
\usepackage{amsmath} % for 'cases' env. and '\text' macro
\newcommand\tfij{\mathit{tf}_{\!ij}}
\begin{document}
\begin{equation}
\label{eq:13aa}
W_{ij} =
\begin{cases}
\hfill \tfij \times \log(N/n_i)
& \text{if $\phantom{0<{}}AW_i = 0$}\\
\hfill AW_i \times \tfij \times \log(N/n_i)
& \text{if $0<AW_i<1$}\\
(1 + \log AW_i) \times \tfij \times \log(N/n_i)
& \text{if $\phantom{0<{}}AW_i \geq 1$}
\end{cases}
\end{equation}
\end{document}