This is partially an electrical engineering problem, but the actual issue apparently lays in my maths.
I have the equation $\frac{V_T}{I_0} \cdot e^{-V_D\div V_T} = 100$
$V_T$ an $I_0$ are given and I am solving for $V_D$.
These are my steps:
Divide both sides by $\frac{V_T}{I_0}$: $e^{-V_D\div V_T} = 100 \div \frac{V_T}{I_0}$ Simplify: $e^{-V_D\div V_T} = \frac{100\cdot I_0}{V_T}$ Find the natural log of both sides: $-V_D\div V_T = \ln(\frac{100\cdot I_0}{V_T})$ Multiply by $V_T$: $-V_D = V_T \cdot \ln(\frac{100\cdot I_0}{V_T})$ Multiply by $-1$: $V_D = -V_T \cdot \ln(\frac{100\cdot I_0}{V_T})$ Now if I substitute in the numbers; $V_T \approx 0.026$, $I_0 \approx 8 \cdot 10^{-14}$ $V_D = -0.026 \cdot \ln(\frac{8 \cdot 10^{-12}}{0.026})$ Simplify a little more: $ V_D = \frac{26}{1000} \cdot -\ln(\frac{4}{13} \cdot 10^{-9})$ and you finally end up with $V_D \approx 0.569449942$
There is an extra step to the problem as well: the question calls not for $V_D$, but for $V_I$ which is a source voltage and that can basically be determined by solving this: $V_I \cdot \frac{1}{40} = V_D$ I.e. $V_I = 40V_D$; which makes $V_I \approx 22.77799768$.
However, this is off by quite a bit (the answer is apparently $1.5742888791$).
Official Solution:
We find $V_D$ to be $\approx$ 0.57.
(Woo! I did get that portion right.)
Since we know that $\frac{V_D}{i_D}$ is 100, we find $i_D$ to be 0.00026.
Background: $\frac{V_D}{i_D}$ is the resistance that I was originally solving. $V_D$ was the voltage across the element, and $i_D$ was the current through the element.
However, either I'm making some stupid mistake but if $i_D = \frac{V_D}{100}$ then how did they get 0.00026? Completing the rest of the solution's method (quite convoluted in comparison to mine, checked mine though another method; $V_I$ does in fact equal $40V_D$), with the correct value, 0.0057, I arrived at exactly the same final value as before.
Would it be fair to say that it is likely that my logic is correct?