Logarithms

A logarithm is the inverse operation to exponentiation. Given a base and , the logarithm is the exponent you would raise to in order to get .

Logarithm

This is defined as:

Interchange of bases

Proposition

We have the following equality

Note this follows as if we let then . So we have Lets equate this all to and use the definition above, on the left hand side we have and on the right hand side which combining these and substituting for gives

Natural logarithms

A lot of people will only talk about logarithms using the base instead of using the notation they may use the notation . This is sort of implied when people just write . The reason for this is and only differ by a constant , as above showed us - in some fields people don’t care about things up to scalar multiple.

Some people use to mean instead.

You need to know the context of the field you are working in. If you are working in a particular base setting it might mean that base. Like in binary it might mean .

Logs turn multiplication into addition

Logs are useful as they bring the rules of exponential powers to regular terms.

Proposition

We have the following equality

Which is proven by the fact that This by extension gives The inverse also holds

Taylor expansion

Proposition

The Taylor series for is as follows