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area (under the hyperbola or in a hyperbolic sector in standard position). Then the natural logarithm could be recognized as the inverse function to the transcendental exponential function e^x.

Proposition: Given 0 < a < b and P = (a, 1/a), Q = (b, 1/b), the signed area of the hyperbolic sector POQ is log b/a.

:proof: In the figure, POQ = POS + PQRS − QOR. Then the equality

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