User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

ln
Something like this, PM?

Theorem
Let $\phi: y \mapsto \phi \left({y}\right)$ be a linear operator.

Let $\log_a$ be the general logarithm.

Then:


 * $\phi \circ \log_a y = \dfrac 1 {\log_a x} \left({\phi \circ \ln y}\right)$

where:


 * $\ln$ is the natural logarithm


 * $\circ$ is function composition

Proof

 * That's the sort of thing I mean. Strangely I've never actually seen it as a result in any maths book.
 * Except the notation used in real calculus tends not to use $\circ$ for concatenation of functions, it would be $\phi \left({\log_a y}\right)$ etc. $\circ$ is a more modern notation that arose through the study of abstract algebra, when it was realised that composition of mappings could be considered as a binary operation. Real analysis (at least at undergraduate level and below) has not caught up with this yet. --prime mover 16:44, 16 January 2012 (EST),
 * I only use it because I think it looks "cleaner" than too many nested parentheses. --GFauxPas 16:53, 16 January 2012 (EST)