Simplify the Expression

28 October 2013

Celn⁡2x=C(eln⁡x)ln⁡x=Cxln⁡xCe^{\ln^2 x}=C\left(e^{\ln x}\right)^{\ln x}=Cx^{\ln x}

This sequence interests me chiefly because at first glance it looks wrong. eln⁡2xe^{\ln^2 x} should expand to (eln⁡x)2\left(e^{\ln x}\right)^{2}, should it not? Only, it doesn’t because of the algebraic properties of exponents.

xy+xy=2xyxy⋅xy=(xy)2=x2y(xy)y=xy2(x∧y)∈R\begin{array}{rrl} x^y+x^y&&=2x^y\\ x^y\cdot x^y&=\left(x^y\right)^2&=x^{2y}\\ \left(x^y\right)^y&&=x^{y^2}\\ &\left(x\land y\right)\in \mathbb{R} \end{array}

Given such context, it makes more sense and becomes more obvious why the original simplifies the way it does.

I restrict xx and yy to R\mathbb{R} because I haven’t yet investigated whether the properties hold for the imaginary numbers.