Starting with macOS High Sierra (10.13), Apple has a new policy that requires user approval for all KEXT (driver) loading. You may see a "System Extension Blocked" prompt (shown below) during the installation of Logitech Options or Logitech Control Center (LCC). 

If you see this message, you'll need to approve the loading of the KEXT manually so that your device drivers can be loaded and you can continue to use its functionality with our software. To allow KEXT loading, please open System Preferences and navigate to the Security & Privacy section. On the General tab, you should see a message and an Allow button, as shown below. In order to load the drivers, click Allow. You may need to reboot your system so the drivers are properly loaded and the functionality of your mouse is restored.

NOTE: As set by the system, the Allow button is only available for 30 minutes. If it has been longer than that since you installed LCC or Logitech Options, please restart your system to see the Allow button under the Security & Privacy section of System Preferences.
 

NOTE: If you don’t allow KEXT loading, all devices supported by LCC will not be detected by software. For Logitech Options, you need to perform this operation if you're using following devices:

• T651 Rechargeable trackpad
• Solar Keyboard K760
• K811 Bluetooth keyboard
• T630/T631 Touch mouse 
• Bluetooth Mouse M557/M558