Logarithm

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.

• 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?

In particular you here find proofs of the basic properties of $\log(x)$:
• $\log(ab)=\log(a)+\log(b),\quad \log(a^r)=r\log(a)$.