Is there proof show that $\log x$ is undefined and make no sense at $ x=0$? I was asked by someone: why $\log x$ is undefined at $x=0 $?
Is there proof  show that $\log x$ is undefined at $x=0$?
Note(01):: log is the inverse function of the exponential function.
note(02): I edited my question as I meant why it's not make a sens at $x=0$ ?
Thank you for your help .

The natural logarithm is defined as 
$$
\log x = \int_{1}^{x} \frac{1}{t} \, dt
$$
Naively plugging in $0$, we get
$$
\log 0 = \int_{1}^{0} \frac{1}{t} \, dt = - \int_{0}^{1} \frac{1}{t} \, dt,
$$
However, the area under $y = 1 / t$ from $0$ to $1$ diverges to $\infty$, which would imply
$$
\log 0 = - \infty
$$
Intuitively, that's why $\log 0$ is undefined, which is inline with your original question.
