Properties of Natural Logarithm
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Theorem
Let $x \in \R$ be a real number such that $x > 0$.
Let $\ln x$ be the natural logarithm of $x$.
Then:
Natural Logarithm of 1 is 0
- $\ln 1 = 0$
Natural Logarithm of e is 1
- $\ln e = 1$
Logarithm is Continuous
The real natural logarithm function is continuous.
Derivative of Natural Logarithm Function
Let $\ln x$ be the natural logarithm function.
Then:
- $\map {\dfrac \d {\d x} } {\ln x} = \dfrac 1 x$
Logarithm is Strictly Increasing
- $\ln x: x > 0$ is strictly increasing.
Logarithm is Strictly Concave
- $\ln x: x > 0$ strictly concave.
Logarithm Tends to Infinity
- $\ln x \to +\infty$ as $x \to +\infty$
Logarithm Tends to Negative Infinity
- $\ln x \to -\infty$ as $x \to 0^+$