According their site, the mouse works with 'Bluetooth Smart Ready device'.
http://www.logitech.com/en-us/product/mx-anywhere2 ( Check System Requirements in TECHNICAL SPECIFICATIONS section )
- Jul 14, 2020 Logitech MX Master 3 Advanced Wireless Mouse for Mac - Bluetooth/USB MASTERED FOR MAC - MX Master 3 for Mac is optimized for macOS, iPad compatible, and finished in Space Gray. ULTRAFAST MAGSPEED.
- How to download Logitech MX Master Wireless Mouse. Download MX Master Wireless Mouse. Save as on your Laptop or PC. Logitech MX Master Wireless Mouse Download Logitech MX Master Wireless Mouse Software for Windows 10, 8, 7 Logitech Options. Lets you toggle between standard and enhanced key functions on your Mac, displays battery status, and more.
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Bluetooth Smart Ready on Mac means Bluetooth 4.0 hardware with Yosemite or later.
Unfortunately BT on your Macbook Pro (mid 2010) is BT2.1+EDR
Mx Master Mouse Software
Mx Master Mouse Software Mac Pro
( If you want to be sure, Logitech provides a procedure to check the PC/Mac is Bluetooth Smart Ready or not )
Mx Master Mouse Software Mac Pro
I guess your only option is use of USB hub if you don't have available USB port,