Natural logarithm

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red: natural logarithm. Note that the legend is in error in that it confuses the base 2 and base e lines.

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 .