Sum of Logarithms/Natural Logarithm/Proof 3
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Theorem
Let $x, y \in \R$ be strictly positive real numbers.
Then:
- $\ln x + \ln y = \map \ln {x y}$
where $\ln$ denotes the natural logarithm.
Proof
Let $\sequence {f_n}$ be the sequence of mappings $f_n : \R_{>0} \to \R$ defined as:
- $\map {f_n} x = n \paren {\sqrt [n] x - 1}$
Let $\map M t = \max \set {\size {t - 1}, \size {\dfrac {t - 1} t} }$
From Bounds of Natural Logarithm:
- $\forall t \in \R_{>0}: \size {\map {f_n} t} \le \map M t$
Fix $x, y \in \R_{>0}$.
Then:
| \(\ds \size {\map {f_n} {x y} - \paren {\map {f_n} x + \map {f_n} x} }\) | \(=\) | \(\ds \size {n \paren {\sqrt [n] {x y} - 1 - \sqrt [n] x + 1 - \sqrt [n] y + 1} }\) | Definition of $\map {f_n} x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds n \size {\sqrt [n] {x y} - \sqrt [n] x - \sqrt [n] y + 1}\) | Absolute Value Function is Completely Multiplicative | |||||||||||
| \(\ds \) | \(=\) | \(\ds n \size {\sqrt [n] x - 1} \size {\sqrt [n] y - 1}\) | Absolute Value Function is Completely Multiplicative | |||||||||||
| \(\ds \) | \(=\) | \(\ds n \frac {\size {\map {f_n} x} } n \frac {\size {\map {f_n} y} } n\) | Definition of $\map {f_n} x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 n \size {\map {f_n} x} \size {\map {f_n} y}\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \frac {\map M x \map M y} n\) | Inequality of Product of Unequal Numbers | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \size {\map \ln {x y} - \paren {\map \ln x + \map \ln y} }\) | \(=\) | \(\ds \size {\lim_{n \mathop \to \infty} \map {f_n} {x y} - \paren {\lim_{n \mathop \to \infty} \map {f_n} x + \lim_{n \mathop \to \infty} \map {f_n} y} }\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \size {\map {f_n} {x y} - \paren {\map {f_n} x + \map {f_n} y} }\) | Modulus of Limit | |||||||||||
| \(\ds \) | \(\le\) | \(\ds \lim_{n \mathop \to \infty} \frac {\map M x \map M y} n\) | Limit of Bounded Convergent Sequence is Bounded | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map M x \map M y \lim_{n \mathop \to \infty} \frac 1 n\) | Multiple Rule for Real Sequences | |||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) |
Thus:
- $\ds \lim_{n \mathop \to \infty} \map {f_n} {x y} = \lim_{n \mathop \to \infty} \paren {\map {f_n} x + \map {f_n} y}$
Hence the result, from the definition of $\ln$.
$\blacksquare$