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This question may have a simple answer or a very complex one, but I am interested in what the reasons are for logarithms and exponential functions having the properties they have. To my knowledge logarithms are the only function that take multiplication into addition and vice versa for the exponential function. I'm sure there must be functions I'm missing.

My question is: What are the algebraic reasons that these functions have these properties and what role does that play in their ubiquity? In addition, resources for further reading (ideally, to the extent possible, without the needing significant background in algebra) would be much appreciated.

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The short answer is that powers are defined so that e.g. $a^x \cdot a^y = a^{x + y}$, whatever $a \ne 0$ and whatever $x$ and $y$ might be. Then the inverse function, $\log_a x$, is easy to dream up: it is, again by definition, $a^{\log_a x} = x$. From there, and the property of powers, you have that $\log_a x + \log_a y = \log_a (x y)$. There are no other functions with these precise properties.

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