The logarithmic Mahler measure \(m(f)\) of a rational function of \(n\) variables is defined to be the average over the \(n\)-torus of \(\log | f| \). The authors also consider a \(q\)-analogue of this where \(\log\) is replaced by the \(q\)-logarithm. Most of the paper concerns \(m(1 + \lambda h(x,y))\), where \(h(x,y) = x + 1/x + y + 1/y\) and \(\lambda\) is a parameter that they assume is real. They prove that if \(0 \leq \lambda < 1/4\) then \(m(1+\lambda h) = \int_0^{4\lambda} (1 - (2/\pi)K(t))\,dt/t\). Here \(K(k)\) is the elliptic integral of the first kind. For \(\lambda > 1/4\), they prove that \(m(1+\lambda h) = \int_0^{1/(4 \lambda)} (2/\pi)K(t) \,dt + \log \lambda\). From this they deduce the interesting functional equation \(2m(h+a+b) = m(h+a^2) + m(h+b^2)\) for \(a, b > 0\), with \(ab = 4\). They point out that the special case \(a = 1, b = 4\) shows the equivalence of two identities conjectured by the reviewer. Since \(m(h + z)\) is an analytic function of \(z\) for complex \(z\) outside the real interval \([-4,4]\), it is clear that the identity has a larger range of validity than is proved in the paper.