  This paper focuses on the problem of finding a continuous extension of the
hypercomplex logarithm along a path. While a branch of the complex logarithm
can be defined in a small open neighbourhood of a strictly negative real point,
no continuous branch of the hypercomplex logarithm can be defined in any open
set $A\subset \mathbb K\setminus \{0\}$ which contains a strictly negative real
point $x_0$ (here $\mathbb K$ represents the algebra of quaternions or
octonions).
  To overcome these difficulties, we introduced the logarithmic manifold
$\mathscr E_\mathbb K^+$ and then showed that if $q\in\mathbb K,\ q=x+Iy$ then
$E(x+Iy) %= (\exp (x + Iy), Iy) = (\exp x \cos y + I\exp x \sin y, Iy)$ is an
immersion and a diffeomorphism between $\mathbb K$ and $\mathscr E_\mathbb
K^+$.
  In this paper, we consider lifts of paths in $\mathbb K\setminus\{0\}$ to the
logarithmic manifold $\mathscr{E}^+_\mathbb K$; even though $\mathbb K
\setminus \{0\}$ is simply connected, in general, given a path in $\mathbb K
\setminus \{0\}$, the existence of a lift of this path to
$\mathscr{E}^+_\mathbb K$ is not guaranteed. There is an obvious equivalence
between the problem of lifting a path in $\mathbb K \setminus \{0\}$ and the
one of finding a continuation of the hypercomplex logarithm $\log_{\mathbb K}$
along this path.
