Max Likelihood problems

1. I have this maximum likelihood problem. When I maximize the
likelihood function, I get some results; when I take the log of the
likelihood function and maximize it, I get different results. I
suppose this is due to the other problem I seem to have, which is too
many local maxima. Is this the typical case? Or is there something
fundamentally wrong in the way I am posing the problem?

2. Econometricians and statisticians like to use log-likelihood
functions instead of likelihood functions. The official argument is
that "it is easier to manipulate/optimize". Yes, it is easier to
derive, for humans. But it transforms a function that is bounded into
something unbounded. If the optimization is going to be made by a
computer does that really help or does that make it more difficult for
the computer/algorithm? I want to do whatever is the more reliable
technique. I hope this is not one of those "case-by-case" things.

Some points:

1. Taking the log can make the likelihood function better behaved. Some of the very large values that can be formed in sub-expressions can often be prevented this way.
2. Many good NLP solvers allow for bounds on the variables that allow you to steer the NLP solver away from the regions where the function can not be evaluated.
3. Use an alternative method to find a good starting point, e.g. a method of moments estimate.
4. Some of these problems can be difficult. Mixture models are a good example.