One reason is for ease of optimizing:
Why to optimize max log probability instead of probability
Another reason relates to better handling of very small values (See Luca's answer):
I am wondering why we use negative (log) likelihood sometimes?
And finally (Courtesy of Whuber's informative comment here: Why apply log to likelihood?) because of the Fisher Information:
The Fisher information is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ upon which the probability of X depends. The probability function for X, which is also the likelihood function for θ, is a function f(X; θ); it is the probability mass (or probability density) of the random variable X conditional on the value of θ. The partial derivative with respect to θ of the natural logarithm of the likelihood function is called the score.
Under certain regularity conditions, it can be shown that the first moment of the score (that is, its expected value) is 0... <snip>
The second moment is called the Fisher information:
Moved from comment to answer:
The derivative of the natural logarithm (there's the need for the log) of the likelihood function gives you the score. The second derivative gives you the Information. Information being a non-computational consideration but one of 'trust' in the estimated parameter value.
High curvature in the region around the MLE means high amount of trust in the estimated value. Thus, using the log-likelihood leads us to a mathematical expression that gives us a handle of how well we're estimating the parameter of interest.