Look at the log file in /usr/local/var/log/elasticsearch.log.

TL;DR

If you look at the formula definition with:

brew edit elasticsearch@6

You will find the plist file used for this service, which define the output log as:

#{var}/log/elasticsearch.log

According to the documentation:

#var ⇒ Object

The directory where the formula's variable files should be installed. This directory is not inside the HOMEBREW_CELLAR so it persists across upgrades.

According to the source code:

def var
  HOMEBREW_PREFIX/"var"
end

You can find the HOMEBREW_PREFIX with:

brew --prefix

It is most likely /usr/local. So you will find your log file in /usr/local/var/log/elasticsearch.log.

Look at the log file in /usr/local/var/log/elasticsearch.log.

TL;DR

If you look at the formula definition with:

brew edit elasticsearch@6

You will find the content of the plist file used for this service, which define the output log as:

#{var}/log/elasticsearch.log

According to the documentation:

#var ⇒ Object

The directory where the formula's variable files should be installed. This directory is not inside the HOMEBREW_CELLAR so it persists across upgrades.

According to the source code:

def var
  HOMEBREW_PREFIX/"var"
end

You can find the HOMEBREW_PREFIX with:

brew --prefix

It is most likely /usr/local. So you will find your log file in /usr/local/var/log/elasticsearch.log.