Node.js Paging with Mongodb
I don't have any good feed back on the technical aspects of your api calls, which look fine to me.  Being that this is code review here is my code review.

I would set a variable to your req.query parameter, which will make the code look cleaner.

Your ternaries can be shortened to:

req.query.limit || 25;


An undefined variable throws an Error, an undefined object property is falsy.

Always use === over ==

Parse Int must always have a radix, otherwise we cannot guarantee consistent performance. (https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/parseInt)

parseInt(order, 10) //Assuming this is sent in decimal


Always return res.send while you are using in fine in this context, adding the return is a safety measure for the future, all you need is someone to change the logic, not properly test error cases and you have a double callback that sometimes can be hard to debug.

So that results in:

var listSort = ['time','request'];
var defSort = listSort[0];
var listOrder = ['-1','1'];
var defOrder = listOrder[0];

router.get('/', function(req, res){
   var rQuery = req.query || {};
   var limit = rQuery.limit || 25;
   var page = rQuery.page || 1;

   var sort =  listSort.indexOf(rQuery.sort) === -1 ? defSort :  rQuery.sort;
   var order = listOrder.indexOf(rQuery.order) === -1 ? defOrder : rQuery.order;
   var querySort = {};
   querySort.sort = parseInt(order, 10);

   var timeBegin = isNaN(Number(rQuery.timeBegin)) ? undefined : Number(rQuery.timeBegin);
   var timeEnd = isNaN(Number(rQuery.timeEnd))  ? undefined : Number(rQuery.timeEnd);
   var request = rQuery.request;
   var queryFind = {};

   if(timeBegin || timeEnd){
       queryFind.time={};
       if(timeBegin)
           queryFind.time['$gte']=timeBegin;
       if(timeEnd)
           queryFind.time['$lte']=timeEnd;
   }

   if(request){
       queryFind.request =  new RegExp(request, 'i');
   }

   Log.find(queryFind).sort(querySort)
   .limit(limit).skip(limit*(page-1), function(err, logItems) {
       if (!err) {
         return res.json(logItems);
       } else {
         return res.send({error: err});
       }
   })});