Logarithm vs Exponential

The function u=u(t)=\exp(t) satisfies

  • \frac{du}{dt}=u,\quad u(0)=1,

which we can rewrite as

  • \frac{dt}{du}=\frac{1}{u},

that is t=\log(u), since t=0 for u=1, and we have defined \log(u) as the primitive function of \frac{1}{u} with \log(1)=0.  In other words

  • u=\exp(t) \quad\mbox{if and only if }t=\log(u)

which means that u=\exp(t) and t=\log(u) are inverse functions. We thus have for t,u>0

  • \ln (\exp(t))=t, \quad \exp(\ln(u))=u .

Writing e^t=\exp(t)  we refer to e as the base of the natural logarithm with exponent t=\ln(e^t).

Since by definition

  • \frac{d}{dx}\log(x)=x^{-1}\quad\mbox{for }x>0,

the logarithm \log(x) fills in the missing value p=0 in the list of derivatives of x^p:

  • \frac{d}{dt}x^p=x^{p-1} \quad\mbox{for }p=\pm 1,\pm 2,\pm 3..,

where we changed the name of the variable from t to x.

Note that with p=0, x^p=x^0=1, and \frac{d}{dt}x^0=0\neq x^{-1}.

Domain and Range of The Logarithm

Since the domain of the function x= \exp(t)>0 is R and range R_+, the domain of the inverse function t=\log(x) is R_+ and range R. In particular,

  • -\log(x)=-\int_1^x\frac{1}{y}\, dy=\int_x^1\frac{1}{y}\, dy
increases without bound as x>0 approaches 0.
Title page of John Napier’s logarithm tables.

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 \log(x):

  • \log(ab)=\log(a)+\log(b),\quad \log(a^r)=r\log(a).


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