Construction and definition of the ln function

Why a new function?

We know how to solve e^x = k for k > 0 using the exponential function, but we do not yet know how to express x in terms of k directly. The natural logarithm, denoted by ln, fulfils this need: it is defined as the reciprocal of the exponential function.

Definition

For any strictly positive real number x, ln(x) is the unique real number t such that e^t = x.

In other words, for x > 0 and for any real number y:

ln(x) = y <=> e^y = x

From this, we deduce two fundamental relations, valid respectively for all x > 0 and for all real numbers x:

e^(ln(x)) = x and ln(e^x) = x

Domain of definition

ln(x) is defined only for x > 0. Writing ln(-2) or ln(0) makes no sense: this is the most common pitfall for pupils. Before working with a ln, one must always check that its argument is strictly positive.

Special values

x	ln(x)
1	0
e	1
1/e	-1
e^2	2

In particular, note that ln(1) = 0 (since e^0 = 1) and ln(e) = 1 (since e^1 = e). These two values serve as reference points in almost all exercises.

Representative graph

The graph of ln is symmetrical to that of the exponential function about the line y = x, since they are reciprocal functions of one another. It passes through the point (1, 0), approaches the y-axis as x tends towards 0 (vertical asymptote) and increases indefinitely, but very slowly, as x tends towards positive infinity.