Say I have data as $x_1,x_2,..x_n$ and I am fitting a model to these data for example, $Y(t)=Y_0 e^{-\lambda t}$. I am trying to estimate the parameter $\lambda$ by fitting to these data.
In these estimation procedures I don't quite understand the meaning of using the logarithm of the data for parameter estimation?
Is there an advantage in using log data when the data vary over a large range, for example from $10^7$ to $1$
When using data without transformation,
model = @(x,t) data(1)*exp(x(1).*t); [lambda,error]=lsqcurvefit(model,initial,t,data);
If using log data I will call the function as,
model = @(x,t) log10(data(1))*exp(x(1).*t); [lambda,error]=lsqcurvefit(model,initial,t,log10(data));
So, the model itself does not change and only the logarithm of the data is taken.
Is there a relationship between the $\lambda$ value that I obtain from the above 2 methods?
Also, which one is actually the correct $\lambda$ value?
When using the log data, should the entire model also be transformed, so that I am fitting to $log(Y)=log(Y_0)-\lambda t$ or can I simply use log of data while the model remains unchanged?