What is $\ln 0$ ?
Is it $-\infty$ or indeterminate?
$\endgroup$ 22 Answers
$\begingroup$$\ln 0$ is undefined. Why is that? Remember that $y = \ln x$ is defined as the unique number staisfying $e^y = x$. But we know that the exponential function is always positive, so what happens if we take $x = 0$? Then there's no $y$ that will make the equation $e^y = 0$ true, so $\ln 0$ is undefined.
However, we can take limits, and in that context we can say $\lim\limits_{x \to 0^+} \ln x = -\infty$, because the closer $x$ gets to $0$ the lower the $y$ we need to take to make $e^y = x$ true. But this does not mean that $\ln 0 = -\infty$! Infinity is not a number and $\ln 0$ doesn't exist, so it doesn't make sense to use them in an equation like this.
$\endgroup$ $\begingroup$The expression $x = \ln 0$ is basically the equivalent to
$$e^x = 0$$
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