log(y) = b* log(x)
b represents the percentage change in y that is associated with a 1% change in x. But this transformation is not always a good idea.
I frequently see papers that examine the effect of temperature (or control for it because they care about some other factor) and use log(temperature) as an independent variable. This is a bad idea because a 1% change in temperature is an ambiguous value.
Imagine an author estimates
log(Y) = b*log(temperature)
and obtains the estimate b = 1. The author reports that a 1% change in temperature leads to a 1% change in Y. I have seen this done many times.
Now an American reader wants to apply this estimate to some hypothetical scenario where the temperature changes from 75 Fahrenheit (F) to 80 F. She computes the change in the independent variable D:
DAmerican = log(80)-log(75) = 0.065
and concludes that because temperature is changing 6.5%, then Y also changes 6.5% (since 0.065*b = 0.065*1 = 0.065).
But now imagine that a Canadian reader wants to do the same thing. Canadians use the metric system, so they measure temperature in Celsius (C) rather than Fahrenheit. Because 80F = 26.67C and 75F = 23.89C, the Canadian computes
DCanadian = log(26.67)-log(23.89) = 0.110
and concludes that Y increases 11%.
Finally, a physicist tries to compute the same change in Y, but physicists use Kelvin (K) and 80F = 299.82K and 75F = 297.04K, so she uses
Dphysicist = log(299.82) - log(297.04) = 0.009
and concludes that Y increases by a measly 0.9%.
What happened? Usually we like the log transformation because it makes units irrelevant. But here changes in units dramatically changed the predication of this model, causing it to range from 0.9% to 11%!
The answer is that the log transformation is a bad idea when the value x = 0 is not anchored to a unique [physical] interpretation. When we change from Fahrenheit to Celsius to Kelvin, we change the meaning of "zero temperature" since 0 F does not equal 0 C which does not equal 0 K. This causes a 1% change in F to not have the same meaning as a 1% change in C or K. The log transformation is robust to a rescaling of units but not to a recentering of units.
For comparison, log(rainfall) is an okay measure to use as an independent variable, since zero rainfall is always the same, regardless of whether one uses inches, millimeters or Smoots to measure rainfall.