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text: 'Welcome to the Arbital introduction to logarithms! In modern education, logarithms are often mentioned but rarely motivated. At best, students are told that logarithms are just a tool for inverting [4ts exponentials]. At worst, they're told a bunch of properties of the logarithm that they're expected to memorize, just because. The goal of this tutorial is to explore what logarithms are actually doing, and help you build an intuition for how they work.\n\nFor example, one motivation we will explore is this: Logarithms measure how long a number is when you write it down, for a generalized notion of "length" that allows fractional lengths. The number 139 is three digits long:\n\n$$\\underbrace{139}_\\text{3 digits}$$\n\nand the logarithm (base 10) of 139 is pretty close to 3. It's actually closer to 2 than it is to 3, because 139 is closer to the largest 2-digit number than it is to the largest 3-digit number. Specifically, $\\log_{10}(139) \\approx 2.14$. We can interpret this as saying "139 is three digits long in [-4sl], but it's not really using its third digit to the fullest extent."\n\nYou might be thinking "Wait, what do you mean it's closer to 2 digits than it is to 3? It plainly takes three digits: '1', '3', and '9'. What does it mean to say that 139 is 'almost' a 2-digit number?"\n\nYou might also be wondering what it means to say that a number is "two and a half digits long," and you might be surprised that it is 316 (rather than 500) that is most naturally seen as 2.5 digits long. Why? What does that mean?\n\nThese questions and others will be answered throughout the tutorial, as we explore what logarithms actually do.\n\n%box:\nThis path contains 9 pages:\n\n1. [40j What is a logarithm?]\n2. [416 Log as generalized length]\n3. [427 Exchange rates between digits]\n4. [44l Fractional digits]\n5. [45q Log as the change in the cost of communicating]\n6. [47k The characteristic of the logarithm]\n7. [4gp The log lattice]\n8. [4h0 Life in logspace]\n9. [4h2 The End (of the basic log tutorial)]\n%',
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