Some community management activities require more time as the number of members increases, others don't. The challenge is identifying these activities and allocating your time accordingly.
The time spent on strategy, business integration, content and, to a lesser extent, growth* is not directly connected to the number of members. These activities will take as much time in a community of 500 members as 5000 members.
However, the time spent on moderation, relationships, technology and, to a lesser extent, activities** will increase as the number of members increases.
This has implications. First, you can easily schedule for the community management activities not affected by the number of members. You can and should spend the same amount of time on strategy, business integration, content and growth every month. You can undertake these activities at the same time.
Second, the time spent on moderation, relationships, activites and technology must grow as the community grows; but not at the expense of the fixed-time activities. This requires strategy. The organization must either hire extra help (volunteers are great, but also require management) or decide what's most important to them.
In a smaller community, more time might be spent on initiating discussions and building relationships. In a larger community, the time might shift to managing volunteers, removing bad content and improving the technology. This isn't the ideal state, it's a trade-off.
Every trade-off will hurt, but it's better to proactively decide your trade-offs than react to what's happening on any given day.
*Just because a community has more members does not mean more time should be spent on growth.
**Activities are similar to content, with the exception that larger activities/events may be needed in a bigger community. Thus, there is a link, but it's not a direct correlation.