I want to plot a particle size distribution.
A typical way to do this is plotting a probability density funtion f(d) over the diameter d. The unit of d is [m], and the unit of f is [1/m]. The area under this curve equals 1.
However if the x-Axes of the graph has a logarithmic scale, the unit of the y-Axis needs to be adapted to ensure that the area under the curve remains 1.
How to correctly label these axis, e.g.:
- Option 1: x:
log(d), [m], y:f(d), [1/log(m)] - Option 2: x:
log(d), [log(m)], y:f(log(d)), [1/log(m)] - Option 3: x:
log(d / m), [1], y:f(log(d / m)), [1] - Option 4: x:
log(d), [log(m)], y:f(d), [1/log(d)]
Option 1 is often seen, but I have doubts if taking a logaritmus of a unit log(m) is allowed. Option 2 is a variant of option 1. In option 3, I'm avoiding the problem by making the argument of the logarithums dimensionless. However this is not very elegant in my opinion, since I am mixing variables and dimensions in f(log(d / m)). The last option is sometimes seen in publications, However I think 1/log(d) is just wrong, since no information on the unit of d is given (ft or m).
Questions:
What is the scientific correct labeling of these axis?
Is it correct to state
log (m) = log ( 1000 mm) = 3 + log (mm)?Things get even more weired, if have combination of units. E.g. How to convert
m/log(m)tomm/log(mm)?
No, the arguments of transcendental functions are unitless. A logarithmic scale is not
$$\log x$$ but $$\log\frac xu$$ where $u$ is some reference magnitude, in the same unit as $x$.