The web site has been updated since the earlier instructions were given.
If you want to view text logs and you are a post-paid customer logged into the desktop version of My Verizon as Account Owner/Manager ... hover over Bill at the top, then scroll down a bit and use the drop-down to select the billing cycle you want to view. Scroll up and click the link to View Activity Usage. You should get a page titled Data, Talk, and Text Activity. Scroll down a bit to the Text section, then click View next to the line's text log you want to view. While bills are available for the past 18 months, text logs are only available for the past 90 days.
To view text logs for the current billing cycle, click Overview at the top of the page, then click View Usage, then click View Messages Details.
To view your text messages, click Plan at the top of the page, then hover over My Plan & Services and click Send a Message.