Why darwins, a measure of evolutionary change, is calculated on the log scale?
There are several purposes for log transformations, but one of the most common that occurs in biology is when changes are relative aka multiplicative. See for example this answer from CrossValidated.

If I told you that one lizard has a tail 5 cm longer than another lizard, that doesn't actually tell you much about how different it is. If the first lizard had a tail 1 cm long, that's a pretty massive difference! If the first lizard had a tail that was 100 cm long, the 5 cm difference is pretty minor.

Taking the log shows this:

Big (relative) change:

$log_e(6) - log_e(1) = 1.8$

$6 - 1 = 5$

Small (relative) change:

$log_e(105) - log_e(100) = 0.049$

$105 - 100 = 5$ 

The simple differences are both 5, but the log differences show you a big change versus a small one.

Logarithms are also more convenient than ratios for rates of change because they are easy to manipulate to predict change over any time interval assuming a constant relative change over time.

Taking the 6cm vs 1cm example above, lets say we want to guess the change when only half of the time is elapsed. If we take half of 1.8 = 0.9 and solve the following for X:

$log_e(X) - log_e(1) = 0.9$

$X \approx 2.5$

So we'd expect after only half the time, the length would have been about 2.5 cm. 

$2.5/1 \approx 6/2.5$ meaning that the relative change for each subdivision of time is equal.

You could try this for any dt, too, and plot out X as a function of time with any dt when you use logarithms. You can't do this directly with a ratio like "6/1" because you can't directly subdivide that "6/1" into smaller time intervals (trying to do so would take you into the world of logarithms again :) ). 

test &lt;- function(time,x1,x2) { 
  out = exp(log(x1)+(log(x2)-log(x1))*time) 
  return(out) } 
time = seq(0,2,0.01) 
plot(time,test(time,1,6),
     xlab = "Time", 
     ylab = "X", 
     type = "l", 
     main = "Change in morphology through time") 
points(x = 1,
       y = 6, 
       pch = 21, 
       bg = "black", 
       col = "black")




Ultimately, what matters most is what thing you are measuring and how you want to interpret a particular type of change. For things like lengths or average counts, often a relative multiplicative change makes the most sense.