# Relationship between Laplace and Legendre transform

In statistical thermodynamics, the canonical, $Z(\beta)$, and microcanonical, $\Omega(E)$, partition functions are related by Laplace tranform. $$Z(\beta) = \int_0^\infty\text dE \ \Omega(E)\ \text e^{-\beta E}$$ The thermodynamic variables corresponding to both ensembles are obtained as logarithm of their respective partition functions. $$S = k_B\ln[\Omega(E)] \qquad -\frac{F}{T} = k_B\ln[Z(\beta)]$$

Now, the quantities $S$ and $-F/T$ are Legendre transforms of each other.

A logarithmic map on Laplace transform pairs gives a legendre transform pair. What is the underlying reason for this ? I have learnt from this question that integral transforms are like basis transformation. Now with regards to this question, where does this interpretation take us ?