Unfortunately you have not supplied the MongoDB logs so again we can’t help with why the service is having issues starting up. I have put steps to get the log data in my original post. Once you post the log then we can provide more help.
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Looking at the screenshot you provided I can see (about half way down) where it says Waiting for connections and then towards the bottom of the screen I can see a connection was made with the MongoDB Shell.
Are you have issues?
Actually the first issue is in brew services it’s stating error with mongoDB community@5.0
And the second is temporary the connections are made through mongo shell and later it’s not able to connect to the server
If brew services start mongodb-community is failing to start it sounds like there is an instance of mongod already running, but that’s just a suggestion from what I’ve seen in the past. Again, the screenshot only shows a limited amount of the log file and from what I can see the mongod service is indeed running.
What are the results of running ps -ef | grep mongod? If you get a result then kill the process, delete the current log file and try staring via brew once more. If mongod fails to start the log should be relatively small. and it will be easier to see what’s happening.
OK, so this shows that mongod is running and explains why you can connect via either mongo or mongosh. It also explains why you are not able to start the process using brew start ....
The 0 in the first column of the first line show that is running under the root user which is not good. I would run kill 334 to stop that process and then brew service start mongodb-community to allow brew to run the process.
One thing to note here is that since the process is currently running as root, that user probably owns the directories and files and those will need to be either deleted or have their ownership changed to your normal user so brew start ... does not fail due to permission issues.
I can’t stress this enough, you should never run a service/daemon as the root user unless explicitly told to do so, and even then only do it as long as you understand the risks involved with doing so.
@Doug_Duncan sir I did same as you said and getting this
Did you check permissions/ownership of dbpath & logpath directories as Doug_Duncan mentioned above?
Since it was run as root previously they might be owned by root and when you start the service as normal user mongod is not able to write
Your log shows only shutdown related messages none related to startup
Please show full path by pwd command
Did you try restart of your service
Also can you spinup another mongod on different port,dbpath,logpath where mongod can write
From command line
mongod --port 29000 --dbpath your_homedir --logpath your_homedir/mongod.log --fork
If it comes up connect by issuing
mongo --port 29000
Please refer to mongo documentation for various command line params
Ok sir let me try and get back to you
@Doug_Duncan sir or @Ramachandra_Tummala sir
kindly refer to this screenshot
This means your mognod process is running and you have successfully connected to it.
Enabling authentication will get rid of the first message.
The second message is informational and I believe that the next time that you connect that message will not show.
The third message just states that you have the mongorc.js file which was created by the older mongo tool. As the note states, you can rename that file to mongosh.js and then that message should go away as well.
Really all of these are just informational messages, although on a production system you will want to enable authentication, but on a test system running on your local machine that will no access to the outside and will not be holding sensitive data, leaving authentication turned off is fine.
From what I can see you have a fully functioning system now. Congratulations! 
Thank you, sir, for your help and support if in the future I face any problem I can come back to you right?
Thank you @Doug_Duncan sir and @Ramachandra_Tummala sir