# 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(
*a**b*) = 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(*a**b*) = ln(*e*^{x}*e*^{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}) =*p*ln(*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}) = *n*ln(*x*^{1 / n}). Thus, *p*ln(*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.