Entropy

Because there is a limiting process that minimizes something (in this case the wasted heat), we can define a new state variable based on this limiting idealization. This lets us define a new and very important state function called the entropy $S$. $S$ is necessarily extensive thermodynamic quantities.

Entropy also has quite a different looking definition in statistical thermodynamics--it says that entropy is proportional to the logarithm of the number of states accessible at the systems energy. Because the number of states of two systems together is the product of the number of each system taken individually and because entropy, being an extensive function, must go like the sum of the individual system entropies, entropy must go like a log.