We consider the space $\mathfrak B^1_{\log^\alpha}$, of analytic functions on the unit disk $\mathbb D$, defined by the requirement $\int_\mathbb D|f'(z)|\phi(|z|)\,dA(z)<\infty$, where $\phi(r)=\log^\alpha(1/(1-r))$ and show that it is a predual of the ``$\log^\alpha$-Bloch'' space and the dual of the corresponding little Bloch space. We prove that a function $f(z)=\sum_{n=0}^\infty a_nz^n$ with $a_n\downarrow 0$ is in $\mathfrak B^1_{\log^\alpha}$ iff $\sum_{n=0}^\infty \log^\alpha(n+2)/(n+1)<\infty$ and apply this to obtain a criterion for membership of the Libera transform of a function with positive coefficients in $\mathfrak B^1_{\log^\alpha}$. Some properties of the Cesàro and the Libera operator are considered as well.