# Natural logarithm

red: natural logarithm

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

The natural logarithm has some interesting properties that follow from the multiplicative properties of ex. 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 = ex and b = ey. It follows that ln(ab) = ln(exey) = ln(ex + y). By definition, ln(ex + y) = x + y = ln(ex) + ln(ey). This last expression, of course, is ln(a) + ln(b).

• ln(xp) = 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 $p = \frac{1}{n}$, note that the statement follows by observing that ln(x) = ln((x1 / n)n) = nln(x1 / n). Thus, pln(x) = ln(xp) for all rational numbers p. The statement must therefore hold for all reals p by continuity.

• ln(1) = 0

Proof: $\ln(1) = \ln(1\cdot 1) = \ln(1)+\ln(1)$, whence we must have ln(1) = 0.