In a research paper (available here) that I'm reading, there is a description of the behavior of a photo resistor used in the construction of a more complex sensor, that goes like this :
Its resistance $R_p$ ($\text{k}\Omega$) changes along with the light illuminance $I$ ($\text{lux}$) by a $\log − \log$ relation (Fig. 4(b)) as
$$\log R_p = b \log \ I + \log a$$
with approximate values of $a = 57.55$ and $b = −0.69$.
In the paper, Fig. 4(b) show the following diagram:
The diagram gives the behavior of $\log R_p$ as a function of the illuminance $I$ taking in to account the device tolerances, as given by the producer: precisely the gray band show (in logarithmic coordinates) the limit values $$ R_{p_\text{mean}} \pm n\sigma_{ R_p} $$ that can be assumed by the resistance of a single device from any production lot, where
- $R_{p_\text{mean}}$ is the mean value of the statistical distribution of the value $R_{p}$,
- $\sigma_{R_p}$ is the standard deviation of $R_{p}$, while
- $n\in\Bbb N$ is a sufficiently large integer, meaning that the probability $P\big[R_p-R_{p_\text{mean}} \notin [-n\sigma_{R_p}, +\sigma_{R_p}]\big]$ is sufficiently small (if ${R_p}$ is assumed to have a gaussian distribution, it is customarily assumed that $n\ge 1$)
What I would like understand is how, from the data represented Fig. 4(b), the authors could get the above model formula for ${R_p}$ and the values of $a = 57.55$ and $b = −0.69$.
Basically , the Author might be saying like this :
The Experimental Curve was Initially given in "raw" terms where it was not easy to see the Empirical relation. Converting to "$log$ terms in $x$ Axis" or "$log$ terms in $Y$ Axis" still did not reveal the relation.
When the Converting to "$log$ terms in Both Axes" (that makes it $\log-\log$ Scale) , the relation was linear $y=mx+c$ , where $y=\log(r)$ & $x=\log(I)$.
Standard regression & curve-fitting gave the linear relation values $m$ & $c$.
I have written the $\log$ values on the Axes.
We can see that the green lines are very good approximations to the grey region.
Out of all the green lines , using "Best-fit" or other Math Criteria or even techniques within the Experimental Physics Community (or Photo-resistor Sub-Community) , the "Best" values were $m=−0.69$ & $c=10^{57.55}$.
In Photo-resistor terms , $b=m=−0.69$ & $a=\log(c)=\log(10^{57.55})=57.55$
[[ I am only answering the Thought-Process & the Math Technique , I am not a Physicist & hence I can not choose the "Best" & more-over, I have not verified the Claims & the Calculations made by the Author ]]