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This paper focuses on the problem of finding a continuous extension of thehypercomplex logarithm along a path. While a branch of the complex logarithmcan 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 openset $A\subset \mathbb K\setminus \{0\}$ which contains a strictly negative realpoint $x_0$ (here $\mathbb K$ represents the algebra of quaternions oroctonions).  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 animmersion and a diffeomorphism between $\mathbb K$ and $\mathscr E_\mathbbK^+$.  In this paper, we consider lifts of paths in $\mathbb K\setminus\{0\}$ to thelogarithmic 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 equivalencebetween the problem of lifting a path in $\mathbb K \setminus \{0\}$ and theone of finding a continuation of the hypercomplex logarithm $\log_{\mathbb K}$along this path.

On a continuation of quaternionic and octonionic logarithm along curves and the winding number