That concludes our introductory tutorial on logarithms! You have made it to the end.

Throughout 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:

- $~$\log_b(x) = y$~$ means [416 "it takes about $~$y$~$ digits to write $~$x$~$ in base $~$b$~$."]
- $~$\log_b(x) = y$~$ means [44l "it takes about $~$y$~$ $~$b$~$-digits to emulate an $~$x$~$-digit."]
- $~$\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$~$]
- 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.$~$

For 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).

We'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 very likely a logarithm function, and that, in essence, There is only one logarithm function.

We've glanced at the underlying structure that all logarithm functions tap into, and we've briefly discussed what makes working with logarithms so dang useful.

There 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:

- Why is $~$e$~$ the natural base of the logarithm?
- What is up with the link between logarithms, exponentials, and roots?
- What is the derivative of $~$\log_b(x)$~$ and why is it proportional to $~$\frac{1}{x}$~$?
- How can logarithms be efficiently calculated?
- What happens when we extend logarithms to complex numbers, and why is the result a [-multifunction]?

Answering these questions will require an advanced tutorial on logarithms. Such a thing does not exist yet, but you can help make it happen.