# Simplify the Expression

## 28 October 2013

$Ce^{\ln^2 x}=C\left(e^{\ln x}\right)^{\ln x}=Cx^{\ln x}$

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

$\begin{array}{rrl} x^y+x^y&&=2x^y\\ x^y\cdot x^y&=\left(x^y\right)^2&=x^{2y}\\ \left(x^y\right)^y&&=x^{y^2}\\ &\left(x\land y\right)\in \mathbb{R} \end{array}$

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

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