Natural logarithm

From Conservapedia

The natural logarithm, ln(x) is the inverse of the function e^x. In other words, if y = e^x, we define ln(y) = x.

The natural logarithm has some interesting properties that follow from the multiplicative properties of e^x. The natural logarithm is also particularly useful in calculating interest.

Properties of the Logarithm

• ln(ab) = ln(a) + ln(b) for all positive reals a,b.

Proof: If a,b > 0, we can write a = e^x and b = e^y. It follows that ln(ab) = ln(e^xe^y) = ln(e^{x + y}). By definition, ln(e^{x + y}) = x + y = ln(e^x) + ln(e^y). This last expression, of course, is ln(a) + ln(b).

• ln(x^p) = pln(x)

Proof: If p is a positive integer, this just follows from repeated application of the above-mentioned additive property of the logarithm. For , note that the statement follows by observing that ln(x) = ln((x^{1 / n})ⁿ) = nln(x^{1 / n}). Thus, pln(x) = ln(x^p) for all rational numbers p. The statement must therefore hold for all reals p by continuity.

• ln(1) = 0

Proof: , whence we must have ln(1) = 0.