Combine two equations with linebreaks into subequations

Question

I have two separate equations with line breaks. However, I would like them to be subequations and, thus, numbered as (1a) and (1b).

I have tried all sorts of things (e.g. working with \begin{subequations}\begin{align} ...) but nothing resulted in decent output. Help would be highly appreciated!

\begin{equation}
\label{eq:first1}
\begin{split}
\text{log}(q_{f,g,c,y}) = b_{11}I_{f,g,c,y-1} + b_{12}\text{log(VarP_{c,y}) 
+ c_{11}\text{log}(P_{c,y}) + 
\\c_{12} \text{log}(CVarCF_{f,g,c,y-1}) + \upsilon_c +e_{1,f,g,c,y}
\end{split} 
\end{equation}

\begin{equation}
\label{eq:first2}
\begin{split}
\text{log}(CVarCF_{f,g,c,y})= b_{21}I_{f,g,c,y-1} + 
b_{22}\text{log(VarP_{c,y}) + c_{21}\text{log}(P_{c,y}) + 
\\c_{22}\text{log}(CVarCF_{f,g,c,y-1}) + \upsilon_c +e_{2,f,g,c,y}
\end{split}     
\end{equation}

Answer 1

Nest split in align.

Some notable points:

1. \text{log} should be \log.
2. multiletter variables should be input as \mathit;
3. the + sign should be at the start of the continuation line;
4. a bit of vertical space has been added between the two equations.

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{subequations}
\begin{align}
\label{eq:first1}
\begin{split}
\log(q_{f,g,c,y}) &= b_{11}I_{f,g,c,y-1} + b_{12}\log(\mathit{VarP}_{c,y})
+ c_{11}\log(P_{c,y})
\\&\qquad +c_{12} \log(\mathit{CVarCF}_{f,g,c,y-1}) + \upsilon_c +e_{1,f,g,c,y}
\end{split}
\\[2ex]
\label{eq:first2}
\begin{split}
\log(\mathit{CVarCF}_{f,g,c,y})&= b_{21}I_{f,g,c,y-1} + 
b_{22}\log(\mathit{VarP}_{c,y}) + c_{21}\log(P_{c,y})
\\&\qquad +c_{22}\log(\mathit{CVarCF}_{f,g,c,y-1}) + \upsilon_c +e_{2,f,g,c,y}
\end{split}
\end{align}
\end{subequations}

\end{document}