I am writing a program to calculate three values from three equations. The three equations are based on one equation with variables substituted in each time.
The base equation:
$$O = \frac{\log(e^{vm} + I^p \ell)}{m}$$
We already know three sets of values for $O$, $I$, and $v$. Importantly, $v_1 = 0$ and $O_2 = 0$, allowing us to simplify further. The final equations are: $$\begin{align} O_1 &= \frac{\log(1 + I_1^p\ell)}{m} \\[6pt] 0 &= \frac{\log(e^{v_2m} + I_2^p \ell)}{m} \\[6pt] O_3 &= \frac{\log(e^{v_3m} + I_3^p \ell)}{m} \end{align}$$
How do I algebraically calculate the values of $m$, $p$, and $\ell$?