Which method is NOT used to export data from Salesforce?

The method that does not pertain to exporting data from Salesforce is logging a case with Salesforce. This option does not involve any data manipulation or export processes; instead, it is a support mechanism for users to report issues, seek assistance, or request features from Salesforce.

In contrast, the other choices represent legitimate methods for data export. The Data Export wizard enables users to select specific objects and schedule regular exports of their data. The Data Loader client application is a powerful tool used for bulk data operations, including exporting data from Salesforce into various formats. Using the Data Manager tool is another method that facilitates the management of data, including exporting functionalities.

Each of these valid methods directly involves exporting data, while logging a case does not contribute to exporting any data from Salesforce. Hence, it clearly delineates itself as the correct answer to the question.