Image of Real Natural Logarithm
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Theorem
Let $\ln$ be the natural logarithm function on the real numbers.
Then the image of $\ln$ is the set of real numbers:
- $\Img \ln = \R$
Proof
By definition of natural logarithm:
- $\ln^{-1} = \exp$
From Exponential Tends to Zero and Infinity:
- $\Dom \exp = \openint {-\infty} {+\infty}$
- $\Img \exp = \openint 0 {+\infty}$
From Exponential is Strictly Increasing:
- $\exp$ is strictly increasing.
From Strictly Monotone Real Function is Bijective, $\exp: \R \to \R_{>0}$ is a bijection.
Thus:
- $\Img \ln = \Dom \exp$
and so $\Img \ln = \R$.
$\blacksquare$