I would run that log periodic antenna vertically unless you plan to use SSB "regularly". Most stations 144 MHz and up are vertically polarized, especially using FM. Imagine going from 500 watts to 5 watts, or more, with the wrong polarization. That loss is also during RX as well. You'll need a rotor, or again you will suffer TX/RX loss if not pointed in the direction of the station you wish to TX/RX. If I had to use only one antenna and needed that bandwidth, I would just use a discone and enjoy the omni-directional aspect without having to constantly spin the antenna. Just put the discone above the roofline and enjoy the significantly easier mounting aspect, versus what would be needed for that particular log periodic. ( I use multiple antennas that offer gain on all the frequencies you mentioned - I'm in no mood for one antenna, but that's me )
Also, a 125' run of coax is not going to be friendly on UHF unless you plan on running some decent coaxial cable like LDF4-50A, LMR-600, or better. If you could do a 50' run even LMR-400 coax would be okay.