My first question is:
What is the proper definition of logarithmic function $f(z)=\ln{z}$.
where $z\in \mathbb{C}$.
quoting Wikipedia.
a complex logarithm function is an "inverse" of the complex
exponential function, just as the natural logarithm $\ln{x}$ is the
inverse of the real exponential function $e^x$.
In a book Calculus Vol 1.By Tom M. Apostol, he tells the function
$\ln{x}$ is defined as $\ln{x}=\int_{1}^{x}{\frac1t\;dt}$ $\color{blue}{\star}$
and the function
$e^x$ is defined to be it's inverse
(rather than the opposite).
Some reasons why it is so as per the book and what I have understood is .
We can define what is $e^2$ $=$ $e\times e$ . But how can we give such a defintion to $e^{\sqrt{2}}$ or$\large e^{\sqrt{2+\sqrt[3]{3+\sqrt[5]{5}}}}$ or more generally the function $a^x$ when the domain is $\mathbb{R}$. Hence as the funciton $a^x$ is not properly defined for Real domain, how can we think about it's definition of it's inverse(The way Wikipedia and some other books define natural logarithm)
So if we are to define $\ln{x} $ as in $\color{blue}{\star}$, it solves all the problem( a proper definition of $\ln{x}$ in real domain , a definition for exponential function in real domain, getting rid of otherwise-circular proofs of some basic theorem in limits involving logarithm and exponential function)
Thinking in the same way just like $e^z$ have problems with definition.How can we define it's inverse?.
Doing a bit of research through internet I found some bits of information.
Wikipedia entry: Natural Logarithm says:
The first mention of the natural logarithm was by Nicholas Mercator in
his work Logarithmotechnia published in 1668,2 although the
mathematics teacher John Speidell had already in $\color{red}{1619}$ compiled a table
on the natural logarithm.[3] It was formerly also called hyperbolic
logarithm,[4] as it corresponds to the area under a hyperbola. It is
also sometimes referred to as the Napierian logarithm, although the
original meaning of this term is slightly different.
Wikipedia entry:Exponential Function
The exponential function arises whenever a quantity grows or decays at
a rate proportional to its current value. One such situation is
continuously compounded interest, and in fact it was this that led
Jacob Bernoulli in $\color{red}{1683}$[4] to the numbernow known as e. Later, in 1697, Johann Bernoulli studied the calculus
of the exponential function
The dates(as per source) of discoveries suggests such($\color{blue}{\star}$) a definition.
So summing up I have two questions.
1.A proper definition of $\ln{z}$ when $z\in \mathbb{C}$
2.Is't the definition for logarithm in real domain, the one I have mentioned ($\color{blue}{\star}$) is the best/correct (Just because I have only seen a few places it is defined so).?
Best Answer
You can define the complex logarithm as $$ Lg(z)=lg(|z|)+iArg(z) $$ where $lg$ is the real logarithm and $Arg(z)$ is the principal branch of the argument (that is, $Arg(z)\in(-\pi ,\pi )$ ). The logarithm is defined for all $z$ in $U=\mathbb{C}-\{x\in\mathbb{R} | x\leq 0\}$.