For Target Display Mode on Mid-2010 iMac, can the Logitech K750 send the right signal to activate it?

I have a new computer with better specs than my iMac 27", mid-2010. I use the target display mode feature for this. However, I wanted to use my Magic Mouse and Apple keyboard on my new computer, and just hook up old Logitech wireless mouse/keyboard for the iMac. However, I later found out that only a wireless Apple keyboard can send the proper signal to make the monitor go into target display mode. Anyone try using the Logitech K750 to send the signal? A little cheaper than the Apple keyboard, and just ***** I have to buy another keyboard and mouse just to "turn on" the monitor.