I had the same thing happen on my Mac. The last line in the log file is "
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Version: Evernote Mac 6.13.1 (455786) [App Store]
Locale: en-US-US
OS: 10.12.5
WebKit: 603.2.4
WebCore: 603.2.4
Username: <not signed in>
Email: <not signed in>
Mac Model: MacBookPro12,1
CPU Type: Intel
CPU Count: 4
CPU Speed: 2.700000 GHz
RAM: 8.000000 GB
Date: 2018-01-05 23:26:32 +0000
2018/01/05 15:25:00:090 E|*** main -[ENAppController loginToAuthorizedAccountsOrAskToLogin]_block_invoke | No auth in keychain. We can not login to USERNAME/NUMBERS