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text: 'That concludes our introductory tutorial on logarithms! You have made it to the end.\n\nThroughout this tutorial, we saw that the logarithm base $b$ of $x$ calculates the number of $b$-factors in $x.$ Hopefully, this claim now means more to you than it once did. We've seen a number of different ways of interpreting what logarithms are doing, including:\n\n- $\\log_b(x) = y$ means [416 "it takes about $y$ digits to write $x$ in base $b$."]\n- $\\log_b(x) = y$ means [44l "it takes about $y$ $b$-digits to emulate an $x$-digit."]\n- $\\log_b(x) = y$ means [45q "if the space of possible messages to send goes up by a factor of $x$, then the cost, in $b$-digits, goes up by a factor of $y$]\n- And, simply, $\\log_b(x) = y$ means that if you start with 1 and grow it by factors of $b$, then after $y$ iterations of this your result will be $x.$\n\nFor example, $\\log_2(100)$ counts the number of doublings that constitute a factor-of-100 increase. (The answer is more than 6 doublings, but slightly less than 7 doublings).\n\nWe've also seen that any function $f$ whose output grows by a constant (that depends on $y$) every time its input grows by a factor of $y$ is [4bz very likely a logarithm function], and that, in essence, [-4gm] function.\n\nWe've glanced at the [4gp underlying structure] that all logarithm functions tap into, and we've briefly discussed [4h0 what makes working with logarithms so dang useful].\n\nThere are also a huge number of questions about, applications for, and extensions of the logarithm that we _didn't_ explore. Those include, but are not limited to:\n\n- Why is $e$ the natural base of the logarithm?\n- What is up with the link between logarithms, exponentials, and roots?\n- What is the derivative of $\\log_b(x)$ and why is it proportional to $\\frac{1}{x}$?\n- How can logarithms be efficiently calculated?\n- What happens when we extend logarithms to complex numbers, and why is the result a [-multifunction]?\n\nAnswering these questions will require an advanced tutorial on logarithms. Such a thing does not exist yet, but you can help make it happen.\n',
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