Logarithm Tends to Negative Infinity

Theorem

Let $x \in \R$ be a real number such that $x > 0$.

Let $\ln x$ be the natural logarithm of $x$.

Then:

$\ln x \to -\infty$ as $x \to 0^+$

Proof

From the definition of natural logarithm:

\(\ds \ln x\)

\(=\)

\(\ds \int_1^x \dfrac 1 t \ \mathrm dt\)

The result follows from Integral of Reciprocal is Divergent.

$\blacksquare$

Also see

• Logarithm Tends to Infinity