Let \(m(P)\) be the (logarithmic) Mahler measure and let \(L(E,s)\) be the \(L\)-function of the elliptic curve \(E\). In this paper, the authors study the values of \(m(k+x+1/x+y+1/y)\) for various values of \(k\). In particular, they prove that \[ m(2+x+1/x+y+1/y)= L'(E_{3\sqrt{2}},0) \] and \[ m(8+x+1/x+y+1/y)=L'(E_{3\sqrt{2}},0). \] Those identities were conjectured by Boyd in 1998. Using some modular equations they also prove the identity \[ \begin{split} m(4/k^{2}+x+1/x+y+1/y)\\=m(2k+2/k+x+1/x+y+1/y)+m(2i(k+1/k)+x+1/x+y+1/y)\end{split} \] for \(|k|<1\) and establish some new transformations for hypergeometric functions.