Some logarithmic behaviour is built into the way the world works and some is a consequence of how we perceive things which span very large ranges.
The simplest chemical and nuclear reactions (first order ones) are, for example, inherently logarithmic. Take any first order process (like radioactive decay). The process has a fixed probability of happening (a radioactive atom or an unstable molecule might have a 10% probability of decaying in a minute). What you observe in the bulk with processes like this is that the rate of the reaction decays exponentially. You start with a rate of say 12 units per minute. After a certain time half the initial material has gone and the rate is now 6 after the same time again the rate is now 3 and so on. This is exponential decay emerging from a very simple process with a fixed probability. A logarithm emerging from a probability, if you prefer. This sort of thing is very common in chemistry and physics.
The other cause of logarithms is human perception. Our perception of light and sound are both inherently logarithmic. This is partially because there is no sensible way to map the range of magnitudes of many natural phenomena to a linear scale. The intensity of sound and light vary over huge orders of magnitude. The Sun is about 15 trillion time brighter than the faintest visible star and it is fairly inconvenient to divide this scale into equal bands. So perception tends to divide the scale logarithmically with each perceived difference representing a fixed ratio not a linear difference (each star magnitude, for example is about 2.5 times brighter than the next lower one, but represents a just-about discernible difference).