Consider the interpretation of logarithms as the cost of communicating a message. Every time the number of possible messages to send doubles, your communication costs increase by the price of a coin, or whatever cheaper storage_medium you have that can communicate one of two messages. It doesn't matter whether the number of possible messages goes from 4 to 8 or whether it goes from 4096 to 8192; in both cases, your costs go up by the price of a coin. It is the factor by which the set grew (or shrank) that affects the cost; not the absolute number of messages added (or removed) from the space of possibilities. If the space of possible messages halves, your costs go down by one coin, regardless of how many possibilities there were before the halving.
Algebraically, writing for the function that measures your costs, and, in general, where we can interpret as the number of possible messages before the increase, as the factor by which the possibilities increased, and as the number of possibilities after the increase.
This is the key characteristic of the logarithm: It says that, when the input goes up by a factor of , the quantity measured goes up by a fixed amount (that depends on ). When you see this pattern, you can bet that is a logarithm function. Thus, whenever something you care about goes up by a fixed amount every time something else doubles, you can measure the thing you care about by taking the logarithm of the growing thing. For example:
Conversely, whenever you see a in an equation, you can deduce that someone wants to measure some sort of thing by counting the number of doublings that another sort of thing has undergone. For example, let's say you see an equation where someone takes the of a relative likelihood. What should you make of this? Well, you should conclude that there is some quantity that someone wants to measure which can be measured in terms of the number of doublings in that likelihood ratio. And indeed there is! It is known as (Bayesian) evidence, and the key idea is that the strength of evidence for a hypothesis over its negation can be measured in terms of updates in favor of over . (For more on this idea, see What is evidence?).
In fact, a given function such that is almost guaranteed to be a logarithm function — modulo a few technicalities.
This puts us in a position where you can derive all the main properties of the logarithm (such as for any ) yourself. Check this box if that's something you're interested in doing.