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:
keep in mind that we restrict to ,
and try to solve for x:
This is a quadratic equation
with ex instead of x as the variable. y will be considered a constant.
So using the quadratic formula, we obtain
we have that
for all x, and since
fails to exceed 1 for some y, we have to discard the solution with the minus sign, so
Read that last sentence again slowly!
We have found out that
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