Definition of ln(x)

Recall that the Fundamental theorem of calculus tells us that if we integrate a continuous function f from 1 to x, we get a new differentiable function that is an antiderivative for f. We can't yet integrate 1/x, so we set

ln(x) = integral of 1/t dt from 1 to x.

In other words, ln(x) is the (signed) area shaded below. Thus the derivative of this new function ln(x) is just 1/x and this is true for all x in (0,∞)

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Questions:
1. What is ln(1)?
2. When is ln(x) = 1?
3. When is ln(x) > 0?
4. When is ln(x) < 0?
5. What is the limit of ln(x) as x ->∞ or as x ->0+?

Jason McCullough, Created with GeoGebra