Logarithms – Using Log Data for Parameter Estimation

Say I have data as a $ x_1, x_2, .. x_n $ and I adapt a model to these data for example, $ Y (t) = Y_0 e ^ {- lambda t} $. I'm trying to estimate the parameter $ lambda $ adjusting to these data.
In these estimation procedures, I do not quite understand what is the use of the logarithmic data for the estimation of parameters?

Does the use of log data have an advantage when the data varies over a wide range, such as $ 10 ^ 7 $ at $ 1 $

  1. When using data without transformation,

        model = @ (x, t) data (1) * exp (x (1). * t);
    
    [lambda,error]= lsqcurvefit (model, initial, t, data);
    
  2. If I use the log data, I'll call the function like,

        model = @ (x, t) log10 (data (1)) * exp (x (1). * t);
    
    [lambda,error]= lsqcurvefit (model, initial, t, log10 (data));  
    

Thus, the model itself does not change and only the logarithm of the data is taken.

Is there a relationship between the $ lambda $ value I get from the 2 methods above?

In addition, which one is the right one $ lambda $ value?

When using log data, should the entire model also be transformed, so that I am able to $ log (Y) = log (Y_0) – lambda t $
or can I just use the data log while the model remains unchanged?