In general, discrete and simple (one transistor or diode) non-linear devices aren't identical to each other, so they vary one to another, and vary their behavior with operating temperature, too.
They can be combined in special ways to gain more predictable behavior, but that is done by trading off their non-linearity to get there.
For example, Michael Wyatt's Cascode Peaking Current Source provides a high-speed current source that achieves a very low temperature coefficient by also taking advantage of the 3333 ppm/C aluminum resistors for 1st order compensation.
Or, more commonly, the use of global negative feedback to achieve predictable voltage gain in audio amplifiers.
It does, however, turn out that non-linear devices can be characterized. There was a period of time where companies spent a lot of money on semiconductor test systems. Tektronix used to have an entire division that was very active in this area. Once you have developed a number of important constants for a specific device, it's non-linear behavior can often be usefully predicted over a range of three to five orders of magnitude. (In one case, a Hamamatsu photodiode used in conjunction with thin film narrow band filter provided seven orders of predictable magnitude photo-response range when held at a fixed temperature.)
If a device is sufficiently characterized, more can be said. If just pulling them out of a box and using them, much less. When making many copies of a specific circuit, it may be prohibitively expensive to characterize each and every non-linear device and make adjustments to each circuit. An exception would be measurement instrumentation where calibration against national or global standard references is required. But in circuits where precision and accuracy isn't the goal (using an LED as an indicator light, for example), non-linear device variations should be managed to provide desired approximate behavior.
Keeping with the diode case, there are these ideas of ever-increasing dynamic ranges of utility:
Assume or measure a fixed voltage drop at some specified operating current in the forward direction. For example, a RED LED might be assumed to have \$V_{_\text{FWD}}=2.0\:\text{V}\$ when operated at \$20\:\text{mA}\$. Or, its \$V_{_\text{FWD}}\$ could be quickly measured (characterized) by applying a current source of \$20\:\text{mA}\$ and measuring its \$V_{_\text{FWD}}\$.
\$V_{_\text{D}}=V_{_\text{FWD}}\$. This model works at the specified operating current.
Assume or measure a fixed voltage drop in the forward direction, with an assumed or measured series resistance. For example, a RED LED might be considered to be \$V_{_\text{FWD}}=1.6\:\text{V}\$ and \$R_{_\text{ON}}=20\:\Omega\$. At \$20\:\text{mA}\$, this works out to a \$2\:\text{V}\$ voltage drop, which is about right. This model is only useful over a small range of currents, say \$20\:\text{mA}\pm 10\:\text{mA}\$. Outside of that range, the model works poorly.
To characterize it, measure the voltage across it using two different currents, say \$V_1=2.14\:\text{V}\$ with a current source set to \$I_1=25\:\text{mA}\$ and say \$V_2=1.87\:\text{V}\$ with a current source set to \$I_1=15\:\text{mA}\$. (Assuming actual operation will be around \$20\:\text{mA}\$.) Then find \$R_{_\text{ON}}=\frac{2.14\:\text{V}-1.87\:\text{V}}{25\:\text{mA}-15\:\text{mA}}=27\:\Omega\$ and \$V_{_\text{FWD}}=2.14\:\text{V}-27\:\Omega\cdot 25\:\text{mA}=1.465\:\text{V}\$.
\$V_{_\text{D}}=V_{_\text{FWD}}+I_{_\text{D}}\cdot R_{_\text{ON}}\$. This model works over a small, specified range of operating currents.
Assume or measure parameters for a simplified Shockley diode model. The model parameters here are the saturation current (\$I_{_\text{SAT}}\$, a fictitious y-axis intercept), an ideality or emission coefficient (\$\eta\$, often taken to be 1, ideally, but may be larger than 1), and the Ohmic resistance (\$R_{_\text{S}}\$.)
To characterize this model requires at least three measurements taken at a single temperature of operation. One of these measurements must be taken at a current that is high enough so that \$R_{_\text{S}}\$ dominates and can be adduced from the data, but this should be pulsed for only a short time so that its temperature doesn't change. The other two measurements should be taken at one order of magnitude less current and then at two or three orders of magnitude still less.
\$V_{_\text{D}}=\eta\cdot V_T \cdot \ln \left(1+\frac{I_{_\text{D}}}{I_{_\text{SAT}}}\right)+I_{_\text{D}}\cdot R_{_\text{S}}\$, where \$V_T=\frac{k\,\cdot\, T}{q}\$ is the thermal voltage and at room temperature will be near \$25\to26\:\text{mV}\$. This model works reasonably well over the included range of measurement currents, typically over a few orders of magnitude, so long as the temperature of operation isn't changed.
However, \$I_{_\text{SAT}}\$ is very highly dependent upon temperature -- on the order of \$\propto T^3\$ -- and signed oppositely to \$V_T\$. So, for prediction over varying temperature requires still more measurement points to characterize a few more constants needed to develop \$I_{_\text{SAT}}\left(T\right)\$.
Obviously, apply given directions in educational settings and ignore what I wrote above if it differs. Otherwise, which of the above makes more practical sense will depend on circumstances.
The first two models listed above are relatively easy and closed solutions can be achieved using basic skills in algebra. The third model can be solved using a few steps of iteration and similar basic skills in algebra; or, directly, using the product-log function (branch-0 of the LambertW) and the ability to manipulate exponential functions.
The third model is important in order to understand log-amps, for example.
