1. The logarithm of the probability of multiple joint probabilities **simplifies to the sum** of the logarithms of the individual probabilities (and the sum rule is easier than the product rule for differentiation)

    $\log \left(\prod_i P(x_i)\right) = \sum_i \log \left( P(x_i)\right)$

  2. The logarithm of a member of the family of **exponential** probability distributions (which includes the ubiquitous normal) is polynomial in the parameters

    $\log\left(\exp\left(-\frac{1}{2}x^2\right)\right) = -\frac{1}{2}x^2$

  3. The latter form is both more **numerically stable** and **symbolically** easier to differentiate than the former.

  4. Last but not least, the logarithm is a **monotonic** transformation that preserves the locations of the extrema (in particular, the estimated parameters in max-likelihood are identical for the original and the log-transformed formulation)