To check if customer is loged in or not you can call Magento\Customer\Model\Session::isLoggedIn() method.

If you want to do that properly in *.phtml files you should use helper inside the template, you will see a ton of examples using the ObjectManager, but you should always bear in mind that it’s not a good coding practice.

So, let’s perform the following steps:

• Create YourCompany\ModuleName\Helper\Data
  
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  <?php namespace YourCompany\ModuleName\Helper;   class Data extends \Magento\Framework\App\Helper\AbstractHelper {     public function __construct(         \Magento\Framework\App\Helper\Context $context,         \Magento\Customer\Model\Session $customerSession     ) {         $this->customerSession = $customerSession;         parent::__construct($context);     }       public function isLoggedIn()     {         return $this->customerSession->isLoggedIn();     } }

• Use the previous Helper, in your phtml as follows:
  
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  <?php $helper = $this->helper('YourCompany\ModuleName\Helper\Data'); ?> <?php if($helper->isLoggedIn()) : ?>     logged in <?php else : ?>     not logged in <?php endif; ?>

  
   

Before trying, remember to re-compile, and clean cache.