The SCS equation for evaluating cumulative infiltration of water is $z=at^b$. We have the following data obtained by experiment and I should find the coefficients $a$ and $b$,$$$$\begin{array}{|c|c|} \hline t(\text{min})& z(\text{cm}) \\ \hline 5&1 \\ \hline 10& 2\\ \hline 15& 2.9\\ \hline 25& 4.4\\ \hline 45& 5.5\\ \hline 60& 7\\ \hline 75& 8.5\\ \hline 90& 10\\ \hline 110& 12\\ \hline 130& 14\\ \hline \end{array}
I should find the coefficients by hand (without using software like Excel). I tried Least Square Method, I should minimize, $$M=\sum_{i=1}^{10} r_i^2\qquad \text{where}\quad r_i=at_i^b-z_i$$So I should have $\dfrac{\partial M}{\partial a}=0$ and $\dfrac{\partial M}{\partial b}=0$
$$\dfrac{\partial M}{\partial a}=0\Rightarrow \dfrac{\partial M}{\partial a}=\dfrac{\partial M}{\partial r_i}\times \dfrac{\partial r_i}{\partial a}=2\sum_{i=1}^{10}(at_i^b-z_i)t_i^b=0 \Rightarrow a\sum_{i=1}^{10}t_i^{2b}=\sum_{i=1}^{10}t_i^bz_i$$ $$\dfrac{\partial M}{\partial b}=0\Rightarrow \dfrac{\partial M}{\partial b}=\dfrac{\partial M}{\partial r_i}\times \dfrac{\partial r_i}{\partial b}=2\sum_{i=1}^{10}(at_i^b-z_i)\times at_i^b \ln b=0 \Rightarrow a\sum_{i=1}^{10}t_i^{2b}=\sum_{i=1}^{10}t_i^bz_i$$But I got the same equation at the end there is still two unkown values and one equation and don't know how to proceed.
Hint: You can convert your problem to a simple linear regression by taking logs, since the relationship $z=at^b$ is equivalent to $$\log z = \log a + b\log t.$$ Now fit a linear regression to a new set of data: $(\log t_1, \log z_1),\ldots,(\log t_{10}, \log z_{10})$. The usual formulas for estimating slope and intercept yield estimates for $b$ and $\log a$. Exponentiate the estimate for the intercept to obtain an estimate for $a$.
As an aside, this is the idea behind plotting data on log-log paper; if you can fit a straight line to the points on a log-log plot, it is straightforward to convert the observed linear relationship into a relationship involving the original (un-logged) variables by reading off the observed slope and intercept and doing a simple calculation.