Logarithms are a frustrating function – the inverse of exponential functions, they are infuriatingly irritating to deal with. But logarithms are a critical part of applied mathematics, and have been a critical tool in the mathematical kit since the dawn of mathematics (Which was when Ugg the caveman wondered how many rocks he had).
Logarithms are what you get when you find what power you must raise ten to to get another number.
The most fascinating part of history is the role of logarithms in the development of arithmetic methods. The simplest aspect is that it is simpler (and less error prone) to add instead of multiplying, so if you find the logarithm of two different numbers, the product of those numbers is equal to the sum of the logarithms. This meant, that in the age before calculators or computers, the process of multiplication was sped up and simplified by using a table of logarithms to do all calculations.
But logarithms are more than just an outdated way of speeding up arithmetic calculations – they are an important tool in graphing. By using a logarithmic scale on one or more axes, you can produce graphs that cover a broad range of numbers by compressing the axis. The graph to the left is an example of this – in one simple picture it presents the entirety of the universe – if this were linear scale, either the small items would be invisible, or the the graph would have to be so large that it is useless.