The reason small changes in a logarithm (like MMS scale) mean big changes in what the logarithm is applied to (like actual earthquake magnitude), is because logarithms count what are called orders of magnitude. A plain-English way to say this is that logarithms tell you how many zeros a number has.
Things in the hundreds have 2 zeros. Things in the millions have 6 zeros. You see how it goes. For this reason, a logarithmic scale can be used to talk about huge ranges, such as the size of the solar system compared to the size of an atom (which is about 23 orders of magnitude in difference).
Logarithms also have properties that we humans often perceive as beauty.