From the SocketAppender documentation
Logging events are automatically buffered by the native TCP
implementation. This means that if the link to server is slow but
still faster than the rate of (log) event production by the client,
the client will not be affected by the slow network connection.
However, if the network connection is slower then the rate of event
production, then the client can only progress at the network rate. In
particular, if the network link to the the server is down, the client
will be blocked.
On the other hand, if the network link is up, but the
server is down, the client will not be blocked when making log
requests but the log events will be lost due to server unavailability.
Since the appender uses the TCP protocol, I would say the log events are "sort of synchronous".
Basically, the appender uses TCP to send the first log event to the server. However, if the network latency is so high that the message has still not been sent by the time a second event is generated, then the second log event will have to wait (and thus block), until the first event is consumed. So yes, it would slow down your application, if the app generates log events faster than the network can pass them on.
As mentioned by @Akhil and @Nikita, JMSAppender or AsyncAppender would be better options if you don't want the performance of your application to be impacted by the network latency.