library mixing log4Esp & Syslog for esp8266
I did a rather dirty (but efficient for my purpose) job : While creating a new logger I declared and attached to it a log4esp formatter in which I sent ... syslog messages 

dirty dirty dirty 

my config constants are : SYSLOG_SERVER, SYSLOG_PORT, DEVICE_HOSTNAME, APP_NAME


My logger.h:

#ifndef Logging_H
#define Logging_H
#include &lt;Log4Esp.h&gt;
namespace Logging
{
void setup() ;
log4Esp::Logger Logger(const char *name);
}
#endif


my logger.cpp: 

#include &lt;Log4Esp.h&gt;
#include &lt;Syslog.h&gt;
#include &lt;WiFiUdp.h&gt;
#include &lt;ESP8266WiFi.h&gt;
#include "Logger.h"
#include "config.h"
namespace Logging
{
// A UDP instance to let us send and receive packets over UDP
WiFiUDP udpClient;
// Create a new syslog instance with LOG_KERN facility
Syslog syslog(udpClient, SYSLOG_SERVER, SYSLOG_PORT, DEVICE_HOSTNAME, APP_NAME, LOG_KERN);
//new logger
log4Esp::Logger Logger(const char *name)
{
  //new instance
  log4Esp::Logger logger = log4Esp::Logger(name);
  //minimal level
  logger.addLevelToAll(log4Esp::Appender::LOG_LEVEL);
  // formatting for serial logging
  logger.addFormatterToAll([name](Print &amp; output, log4Esp::Appender::Level level, const char *msg, va_list * args) {
    ////////////////////////////////////////
    // serial log format
    // output uptime of this program in milliseconds
    output.printf("%10d", millis());
    output.print(log4Esp::Appender::DEFAULT_SEPARATOR);
    // output free heap space
    output.printf("(%d)", ESP.getFreeHeap());
    output.print(log4Esp::Appender::DEFAULT_SEPARATOR);
    // output logger name
    output.printf("%-4s", name);
    output.print(log4Esp::Appender::DEFAULT_SEPARATOR);
    // output log level
    output.print(log4Esp::Appender::toString(level, true));
    if (level &lt;= log4Esp::Appender::Level::ERROR) {
      output.print("***");
    } else {
      output.print("   ");
    }
    output.print(log4Esp::Appender::DEFAULT_SEPARATOR);
    // determine buffer length for formatted data
    size_t length = vsnprintf(NULL, 0, msg, *args) + 1;
    char buffer[length];
    // output formatted data
    vsnprintf(buffer, length, msg, *args);
    output.print(buffer);
    ////////////////////////////////////////
    // Stop here if wifi is down
    if (WiFi.status() != WL_CONNECTED)
      return;
    ////////////////////////////////////////
    // format for syslog
    uint16_t slevel;
    switch (level) {
      case log4Esp::Appender::Level::FATAL    :
        slevel = LOG_CRIT;
        break;
      case log4Esp::Appender::Level::ERROR    :
        slevel = LOG_ERR;
        break;
      case log4Esp::Appender::Level::WARNING  :
        slevel = LOG_WARNING;
        break;
      case log4Esp::Appender::Level::VERBOSE  :
        slevel = LOG_INFO;
        break;
      case log4Esp::Appender::Level::TRACE    :
      default:
        slevel = LOG_DEBUG;
        break;
    }
    syslog.logf(slevel, "%10d|(%d)|%-4s|%s", millis(), ESP.getFreeHeap(), name, buffer);
  });
  return logger;
}
void setup() {

}
}