It's much easier to deal with logarithms, as the relevant numbers are usually very small or very large. If you have a long exponential expression, it's hard to see the difference, but if you're looking at 4.3 vs 5.6, you can immediately see what's happening. And logarithms are a well-known (and well-understood) way of achieving this compression. You can ...
I would like to add details to Oliver's answer.
From the book "Pattern Recognition and Machine Learning" by Bishop (Section 1.2.5):
In practice, it is more convenient to maximize the log of the
likelihood function. Because the logarithm is monotonically increasing
function of its argument, maximization of the log of a function is
equivalent to ...