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As the title suggested, I want to know why people often use log likelihood function, instead of likelihood function by itself. What I know, is that, if $\hat{\theta}$ is the maximum of a likelihood function $f(\theta; \mathbf{x})$, then it is also the maximum of $g(\theta) := \log (f(\theta; \mathbf{x}))$. Are there some other reasons why we are interested in $g$ instead of $f$?

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If you were to use the likelihood function in your maximization routines you may encounter issues of underflow/overflow depending on parameter values and the amount of data you have.

Apart from the numerical reason above, log likelihood functions are sometimes nicer to deal with analytically as product of several factors transforms into a sum and you get rid of the exponential (if you are dealing with normal, exponential densities) which makes life a bit easier when computing MLE analytically.

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