# 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