Meanwhile I got a bit further. The readily made antenna I linked to earlier is a bit large and doesn't have the biconical 'antlers'. After some googling I found this website
https://hamwaves.com/lpda/ which can produce a recipy to make your own log-period antenna. I have used these parameters to get to a trade-off for reasonable gain, not too many dipoles and reasonable size:
INPUT
Lowest frequency f₁ = 200 MHz
Highest frequency fₙ = 1500 MHz
Diameter of the shortest element ⌀ = 6 mm
Characteristic input impedance Zc_in = 50 Ω
Taper τ = 0.880
Optimal relative spacing σₒₚₜ = 0.163
Chosen relative spacing σ = 0.085
And I ended up with this:

Still a bit larger than what I had hoped for though but I wanted to get a decent coverage from 30MHz and onwards. I also found out there is a flaw in the LPDA calculator website. It calculates the total length of the dipoles but that is only for one half of the antenna. I started with buying 8 meters of pipe for the dipoles but ended up needing nearly 13 meters. It is amazing how much length goes into an antenna like this. I used 15x15mm aluminium square tubing for the booms and 6mm round tube for the dipoles. All in all it is pretty light. I soldered the dipoles to the booms using aluminium solder I bought from Harbour Freight (I have visited all kinds of shops in the US while on holiday

). No, this is not my best aluminium soldering job; I put this antenna together in a couple of hours.
For the antlers I used 8mm tube because all shops in the area are out of 2 meter lengths of 6mm tube

I did some comparative testing against the 80cm bow-tie antenna I made a couple of months ago. The log period antenna is way more sensitive. I made this scan from 5MHz to 150MHz and it shows the log-period antenna is 25 dB to 30 dB more sensitive. I used a 27Mc antenna to transmit the test signal coming from the network analyser + amplifier.

The blue (noise floor) and purple (received level) traces are measured using the bow-tie antenna. The yellow (noise floor) and orange (received level) traces are measured using the log period antenna. Probably I can optimise a bit where it comes to directivity but I didn't try that yet.
I have put my new toy to good use already. I needed to measure some emissions from a device in the 900 MHz range and I could pickup the spike between the LTE bands just fine (even though the LTE signal strength is a bit higher). With the bow-tie antenna I needed to use a pre-amplifier in order to get enough signal strength into the spectrum analyser.