An infinite family of homeomorphic, but pairwise non-diffeomorphic smooth compact simply-connected four-dimensional manifolds with the second Betti number 2 bounded  by the Poincare homology sphere is constructed. Also it is constructed an infinite family of spheres embedded into CP2\#2(-CP2) such that each of them has only one point where it is not smooth, and they are ambiently homeomorphic (via homeomorphisms smooth on some their neighbourhoods), but are not ambiently diffeomorphic. It is proved that some pairs of topological logarithmic transformations, which chainge smooth type of CP2#9(-CP2) without changing its topological type, do preserve the smooth type of S^2xS^2.