Since is defined in terms of the exponential function, you should not be surprised that its inverse function can be expressed in terms of the logarithmic function:

Let's set
,
keep in mind that we restrict to ,
and try to solve for *x*:

This is a quadratic equation with

So using the quadratic formula, we obtain

Since we have that for all

and consequently

Read that last sentence again slowly!

We have found out that

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