There are several email logging plugins to choose from for WordPress. We recommend the free and effective option Email Log by Sudar, available at https://wordpress.org/plugins/email-log/. This useful utility doesn’t add anything other than what it’s supposed to, and it doesn’t limit usage to try to sell you a premium version the way several of the other plugin options will.
Long-term if you do want email logging plus other email related features then we recommend WPForm SMTP mailer plugin or Post SMTP.
Sending email directly from your server can often be a hit-or-miss affair. We highly recommend for production sites that you don’t rely on delivery directly from the server without authentication. Using SMTP login with your email service provider can increase delivery rates. If you have GMail services already, it makes sense to route your site outbound email via GMail servers in order to increase the reliability of email delivery.