# Logarithm

The function satisfies

- ,

which we can rewrite as

- ,

that is , since for , and we have defined as the primitive function of with . In other words

which means that and are *inverse functions*. We thus have for

- .

Writing we refer to as the *base of the natural logarithm* with exponent .

Since by definition

- ,

the logarithm fills in the missing value in the list of derivatives of :

- ,

where we changed the name of the variable from to .

Note that with , , and .

# Domain and Range of The Logarithm

Since the domain of the function is and range , the domain of the inverse function is and range . In particular,

increases without bound as approaches 0.

# To Think About

- What was the use of Napier’s logarithms? Are they still used?
- What is a slide rule and how does it work?
- What are the basic rules for computing with logarithms?

# Read More

- \hyperref[log]{The Logarithm.}

In particular you here find proofs of the basic properties of :

- .

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