I just noticed this item from Andrew Sullivan in The Atlantic:
Consider this hypothetical. It's November 2008. A young Pakistani Muslim is watching television and sees that this man — Barack Hussein Obama — is the new face of America. In one simple image, America's soft power has been ratcheted up not a notch, but a logarithm.
Now I've been trying hard to stifle my natural temptation towards mathematico-linguistic pedantry. Really, I have. "A number of" to mean "many" still annoys me — zero is a number; so is one — as does "to the nth degree" (depends on the n, no?). But I have to acknowledge that these are established English idioms, governed by the rules of English idiom, not of mathematics. I don't like 'em, but that's my problem, not the speaker's.
Still, if you're going to try to come up with new figurative usages, it seems to me that the figure of speech should fit rhetorically. "A number of" at least sounds large, but "a logarithm" doesn't. Logarithms, I think, generally seem small. In all the commonly used bases, they are smaller than the original number, often much smaller. A million is a big number; comparatively, the base-ten logarithm of a million (six) is much smaller.
A logarithmic scale does have the property that small steps can correspond to large increases, which is what I take it Sullivan is referring to. But "ratcheted up ... a logarithm" doesn't quite capture that, I think. "Exponential increases" does communicate "large increases," in a way I have to grudgingly accept (down, math pedant self, down!). But logarithm is the opposite of exponential, not a synonym. And when new terms are coined, the correspondence to the original referent does matter, especially given that most people who even know what a logarithmic scale is will likely think of the original referent.
My sense is that scientific allusions, like classical allusions, tempt people into error — they sound cool, and people use them because of that rather than because they're apt. So think twice before you ratchet things up a logarithm.
UPDATE: The winner is commenter Elmer: "To summarize, ratchet and soft and logarithm just don't go together well, though Soft Ratchet Logarithm would, of course, be an OK name for a band."