I have an equation that it seems to be a non-linear equation. I want to compute the parameters a1 till a4.I want to simply do a linear regression to find the parameters, which is much easier than a non-linear regression.
Y = exp (a1 log (x) + a2 log (z) + a3) + a4
Is there a way to linearize it or I have to do a non-linear regression?
On
$$ e^{(a1 log (x) + a2 log (z) + a3)} + a_4$$ $$ =e^{(log (x^{a_1}) + log (z^{a_2}) + a_3)} + a_4$$ $$ =e^{(log (x^{a_1})}e^{+ log (z^{a_2})}e^{a_3)} + a_4$$ $$=x^{a_1}z^{a_2}e^{a_3}+a_4$$ Without $a_4$, you could do a linear regression in the log-transformed variables, but with it there is no way. So it would have to be a non-linear regression.
As Paul said, the model is totally nonlinear with respect to its parameters and then nonlinear regression will be required.
However, admit for the time being that you know $a_4$. So, you can write the model $$y -a_4 = \exp(a_1 \log (x) + a_2 \log (z) + a_3)$$ $$\log(y -a_4) = {a_1 \log (x) + a_2 \log (z) + a_3}$$ So, for given value of $a_4$ you can compute the $y_i ^{calc}$ and then $$SSQ(a_4)=\sum_{i=1}^n \Big(y_i ^{calc}-y_i ^{exp}\Big)^2$$ which is what you want to minimize. Try with different values of $a_4$ until you see more or less a minimum.
In your case, you could continue with the linear regression making a detailed search of the minimum but the problem is that you will be minimizing $$\sum_{i=1}^n \Big(\log(y_i ^{calc}-a_4)-\log(y_i ^{exp}-a_4)\Big)^2$$ while the only thing which has been measured is $y$ and not any of its transforms. Stopping at this point could lead to quite bad results (depending for sure on the noise in the data).
When you get it, you have all the elements required for starting the nonlinear regression (which requires reasonable estimates for starting).
What I described here is a general method which works very well when you have only few parameters making the function nonlinear. I use to do it with even three parameters (building a four dimension grid).