You can use MSGPACK_ADD_ENUM() macro. It has been supported since version 1.0.0. I recommend version 1.1.0 or later.
See:
https://github.com/msgpack/msgpack-c/wiki/v1_1_cpp_adaptor#enums
I applied MSGPACK_ADD_ENUM() to your code:
#ifndef LOG_MSG_HPP_
#define LOG_MSG_HPP_
#include <stdlib.h>
#include <msgpack.hpp>
/** @namespace logd */
namespace logd {
enum class log_level { SILENT,DEBUG };
enum class log_domain { AI, MISC };
class log_msg {
public:
log_msg(log_level lev, log_domain dom, std::string msg):lev_(lev), dom_(dom), msg_(msg) {}
log_level level() { return lev_;}
log_domain domain() { return dom_;}
std::string message() { return msg_; }
~log_msg() {}
MSGPACK_DEFINE(lev_, dom_, msg_);
log_msg() = default;
private:
log_level lev_ {log_level::DEBUG};
log_domain dom_ {log_domain::MISC};
std::string msg_ {"No message given."};
};
} /* namespace logd */
MSGPACK_ADD_ENUM(logd::log_level);
MSGPACK_ADD_ENUM(logd::log_domain);
#endif /* LOG_MSG_HPP_ */
#include <sstream>
#include <cassert>
int main() {
logd::log_msg lm { logd::log_level::SILENT, logd::log_domain::AI, "hello" };
std::stringstream ss;
msgpack::pack(ss, lm);
msgpack::object obj = msgpack::unpack(ss.str().data(), ss.str().size()).get();
logd::log_msg lm2 = obj.as<logd::log_msg>();
assert(lm.level() == lm2.level());
assert(lm.domain() == lm2.domain());
assert(lm.message() == lm2.message());
}