What kind of log is it? Some wood will simply deteriorate rapidly in a terrarium, while the more dense woods such as manzanita, cypress knee, Mt. mahogany, Osage orange, bamboo will grow less mold, support mosses and generally last longer. Cork bark is also very resistant to molds, but grows mosses and such well. Some of the wood you can purchase in a pet store is grape wood, and this really doesn't do very well in a moist environment. I don't know about ghost wood, or even exactly what it is, but I've seen it discussed, and people seem pleased with it. I have found that some decorative items like monkey pods also deteriorate and mold more readily than something like Brazil nut husks or cocoanut "huts."
There is really nothing you can do (except remove the log) that won't affect your other plants and more importantly, the frogs. You are always going to have molds of some sort here and there, but I can assure you that as your tank matures, they will be much less conspicuous.
Here's a thought--Do you have any ferns in your set-ups? Heart fern is particularly easy to do this with: Find a leaf with the brown spores on the underside and scrape some of these off onto your log, or just lay the leaf down on it. The spores will germinate into little green prothalium that look a great deal like liverworts, and eventually sprout new little fern leaves. Or you could lace the log with strands of Java moss to compete with the molds. You can also try to layer the log with long brown sphagnum moss, which will also occasionally, but not always "come alive." and grow a rather pretty green moss.
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Patty
Pahsimeroi, Idaho
4 D. auratus blue
5 D. galactonotus pumpkin orange splash back
7 D. imitator
6 D. leucomelas
4 D. pumilio Bastimentos
4 D. fantasticus
6 P. terribilis mint and organe
4 D. reticulatus
4 D. castaneoticus
2 D. azureus
4 P vittatus
2 P. lugubris