The **natural logarithm**, is the inverse of the function . In other words, if , we define .

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

## Properties of the Logarithm

- for all positive reals .

Proof: If , we can write and . It follows that . By definition, . This last expression, of course, is .

Proof: If 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 . Thus, for all rational numbers . The statement must therefore hold for all reals by continuity.

Proof: , hence we must have .