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One of the earliest ways that the natural logarithm was defined was as the area from 1 to x under -: 1 X Inx - 24 You may recall that logarithmic functions have a product property: In xy = Inc + Ing To see why this property is true using the integral definition, first use properties of integrals to write .cy In xy = dt this way t and in the last integral make the substitution u = ac Changing the limits of integration in the last integral, when t = > we have u and when t = xy we simplify to get u