# Why Logarithms?

It turns out that the only (non-zero differentiable) functions with the
key property
are logarithmic
functions. To see this we derive a differential equation for
whose solution is a logarithm. Differentiating in with
respect to gives
Similarly, differentiating with respect to gives
Hence
In the special case this turns into
From if follows that . The solution of this
initial value problem is
where and is an arbitrary constant.

So only logarithms work. However, if you want to settle for
properties that are more complicated than there are other
possibilities. For example, you may want to think about how to
construct a slide rule based on the identity