Final answer:
A protein kinase cascade features logarithmic growth of the signal during cell signaling and involves phosphorylation and second messengers to amplify the response.
Step-by-step explanation:
The type of amplification that has a logarithmic growth of the signal is a protein kinase cascade. This amplification cascade takes a single effector-receptor interaction and magnifies its effect in the cell by orders of magnitude. The process involves phosphorylation of proteins by kinases and the generation of second messengers like cAMP and calcium ions, which further propagate the signal through a series of relay proteins leading to a substantial cellular response. The kinase cascade results in the activation of large numbers of proteins from a relatively small number of signal molecules, thereby exhibiting a logarithmic growth pattern in signaling.