Logarithmic CFTs have OPEs (and operators) with logarithms. But to have logarithms one needs to have some scale to make the argument of the log a dimensionless quantity. But if the theory has a scale then how can that be a CFT? Standard CFT correlators don't have logs or exponentials precisely for that reason - they don't have any scale.
Standard CFT correlators do have logs since $z^\Delta=\exp (\Delta \log z)$. In logarithmic CFT the only difference is that $\Delta$ is a non-diagonalizable matrix rather than a number. If $\Delta$ is a matrix, the function $z^\Delta$ is still covariant under rescalings, in the sense that $(\lambda z)^\Delta=\lambda^\Delta z^\Delta$. Your logarithmic correlation functions are the matrix elements of $z^\Delta$, and they get mixed with one another by the multiplication with $\lambda^\Delta$.
Conclusion: the terms you get in logarithmic correlators when you do $z\to \lambda z$ are manifestations of the covariance under rescalings.