Generally speaking, you are correct. There's no free lunches. But there are many variables to consider. This is the case when looking at the smaller, compact logs and ones that are poorly designed, or when a Yagi is tuned for max f-b. ... But, boom for boom length, they are within a db of gain and the f-b is similar to a Yagi when the Yagi is tuned for reasonable forward gain. A Yagi tuned for MAX forward gain has a f-b of maybe 15db, so a log may have a better f-b in that case. A yagi tuned for max f-b can exceed 45db over a narrow range, but will lose a ~ db in forward gain, thus back to par with the log. [of similar boom length]
Logs get a bad rap sometimes. I thought the same thing before I looked into them closely. A lot depends upon the design and implementation. There are a few commercial logs out there that use open wire line to inter-connect the els. Big mistake. The eles are ~ 70 ohms and need a good set of 3/8" spaced 1" angle aluminum or a double boom to simulate a 70 ohm feedline. Without that some els will hog RF and others will go along for the ride. Axe Chuck/K1KW about his ideas on this.
As for front to back, mine gets close to 25 db on 20M and sometimes 30db + on 15 and 10M, and the Warc bands. There are times when I turned the log and could not even hear guys who were previously in there with decent signals. Especially on the bands above the lowest design freq, there's a lot of "reflector aluminum" behind the driven el...
As for forward gain, yes there is a compromise, and much depends upon the log's taper and spacing factors. You can put up a 13-30mhz log with a 20' boom and it wud be lucky to perform like a 2el Yagi, poor 12db f-b, etc. BUT, put up one that covers the same freq with a 60'++ boom and you are seeing performance that equates to 4el Yagi monobanders. Chuck/K1KW, who also has a HB stack of 60' logs, says that they are the equiv of a 4-5el Yagi monoband stack - but covers 13-30mhz continuous! But, as you suggested, you can fit a wide spaced 6el monoband Yagi on a 60' boom with somewhat better performance than the log.
Also, consider the fact that logs are virtually unaffected by tower problems, like guy wires, other beams, ant wires, etc. Being a closely spaced, all driven cell, they are not like a Yagi whose performance can be destroyed by surrounding objects. I wud venture to guess that many hams with Yagis that are stacked with other beams and guy wires "think" are working well but are not. It takes very little to ruin a f-b on a sharply tuned Yagi. A log can actually sit 5 ft off the ground and still work with a pattern. The take off angle will be very high, but the elements will still have the proper phase relationship, though pulled down in freq.
BTW, Chuck/KW has been known on the 10-20M bands as having probably the biggest signal and best ears in North America. He gets on and the pile ups are bedlam.
As you can see I love the log periodic - I run a pair of stacked homebrew ones myself. When we consider that there is no element tuning required, the gain and f-b, AND SWR stays within a reasonable limit from 13-30mhz, the gain and f-b rival big monoband Yagis [assuming they are big logs], it's a hard combo to beat.... But takes more effort, more elements, to build a big set of logs - no free lunch.
BTW, modeling a log with just 180-0-180-0, etc feed will not show the full gain. Using Nec and a simulated feedline is closer. According to Chuck, there is a capacitive charging effect on the smaller elements in front of the normal cell that adds to the gain and will not show up in some models. I've never verified this, but thought I'd mention it.
73, Tom, K1JJ
As a footnote: Since writing this I’ve taken down my logs and rebuilt them into stacked monobanders for 20, 15 and 10M. Includes a pair of 60’ long boom 20M Yagis, a pair of 50’ long 15M Yagis and four 24’ boom 5el 10M Yagis. Each band is now optimized for height and gain, but takes up much more space and effort.