Specializing a template member function in a nested template class

Question

Let's say we have a template class with a nested class inside it. The nested class has a template member function. It seems that I managed to declare a specialized version of the member function but cannot define it outside of the class.

Here is an example :

template<class Stream>
class Logger
{
    class LogEntry
    {
    public:
        LogEntry(Logger<Stream>& logger, Stream& stream)
            : m_stream(stream)
        {
        }

        LogEntry(const LogEntry&) = delete;
        LogEntry(LogEntry&&) noexcept = default;

        template<typename Loggable>
        LogEntry& operator<< (const Loggable& loggable);

        template<>
        LogEntry& operator<< <int>(const int& loggable);

    private:
        Stream& m_stream;
    };

public:
    Logger(Stream& stream) : m_stream(stream) {}

    LogEntry operator<<(const auto& loggable)
    {
        LogEntry entry(*this, m_stream);
        entry << loggable;
        return entry;
    }

private:
    Stream& m_stream;
};

// Definition of the previously declared member function (operator)
template<class Stream>
template<typename Loggable>
Logger<Stream>::LogEntry& Logger<Stream>::LogEntry::operator<<(const Loggable& loggable)
{
    m_stream << loggable;
    return *this;
}

// Failing to define specialized version of the function that is successfully declared in the class. With error : "cannot specialize (with 'template<>') a member of an unspecialized template"
template<class Stream>
template<>
Logger<Stream>::LogEntry& Logger<Stream>::LogEntry::operator<<<int>(const int& loggable)
{
    m_stream << loggable;
    return *this;
}

Any idea on what I'm doing wrong in this example?

Thanks in advance