How can I log out of a shared account without the password?

That's a classic scam to hold your account and device for ransom. Your only recourse is likely to be asking Apple to completely reset the device.

Apple can completely reset a device if a person can provide sufficient evidence of valid ownership. Do this online at: Activation Lock support request - https://al-support.apple.com/#/additional-support Apple says, "Proof of ownership documentation is required," but does not detail what currently constitutes proof other than it “must include the product serial number, IMEI or MEID.” Apple may typically require the original receipt* issued by Apple or an Apple Authorized Reseller. If Apple resets the device this will give you access to the device but won't restore access to the account used to lock it. There is nothing that can be done to expedite an Activation Lock request.

*Apple recommends you "Keep your receipt after you buy an Apple device" --> Keep your receipt after you buy an Apple device - Apple Support

Do not share accounts. Make sure that each family member has a unique Apple ID as suggested in this support article --> Make sure that each family member has a unique Apple ID - Apple Support

There is no such thing as a shared account. It sounds like someone talked you into signing in someone else's Apple ID into your iPhone. This will lock your device to their Apple ID.

You may need to provide the original receipt from Apple or Authorized Apple Seller if your device is now locked to someone else's Apple ID to remove the Activation Lock.

How to remove Activation Lock - Apple Support

problemappleid wrote:

how erase?

Follow the instructions in the link I provided about activation lock removal service.

Since you said the other person is the "account owner" I am not sure if going the compromised account path is the one to take. You only do that for your own account. To me you clearly said this was another person's account you were borrowing to get apps you do not yourself have. Doing the password change immediately after somebody has logged into an account is, as I said, a classic scam technique. Unless you have some way you think you can convince Apple this is your account (which you say it is not) then your only recourse is to have the device erased so you can make use of it again.

Limnos wrote:

I believe you are correct. At first I thought the OP signed into another users device with their Apple ID and the other user changed the password which could only be done if the OP gave them the password when it was signed into the device.