  Let $F$ be a non-Archimedean local field of characteristic not equal to 2,
let $E/F$ be a finite unramified extension field, and let $\sigma$ be a
generator of $\text{Gal}(E, F )$. Let $f$ be an element of $Z(H_{I_E})$, the
center of the Iwahori-Hecke algebra for $GL_2(E)$, and let $b$ be the Iwahori
base change homomorphism from $Z(H_{I_E})$ to $Z(H_{I_F})$, the center of the
Iwahori-Hecke algebra for $GL_2(F)$ [8]. This paper proves the matching of the
$\sigma$-twisted orbital integral over $GL_2(E)$ of $f$ with the orbital
integral over $GL_2(F)$ of $bf$. To do so, we compute the orbital and
$\sigma$-twisted orbital integrals of the Bernstein functions $z_{\mu}$. These
integrals are computed by relating them to counting problems on the set of
edges in the building for $SL_2$. Surprisingly, the integrals are found to be
somewhat independent of the conjugacy class over which one is integrating. The
matching of the integrals follows from direct comparison of the results of
these computations. The fundamental lemma proved here is an important
ingredient in the study of Shimura varieties with Iwahori level structure at a
prime $p$ [7].
