= Solution
Parametrize the <logarithmic spiral> by
$$
h:\mathbb R\longrightarrow X,\qquad h(t)=(e^t\cos t,e^t\sin t).
$$
The map is continuous into the ambient plane, hence into $X$ with its <subspace topology>, and it is onto by the definition of the spiral. It is injective because its radius $|h(t)|=e^t$ determines $t$ uniquely, even when the angular coordinate winds through repeated turns.
Every point of $X$ is nonzero. The inverse is therefore
$$
\boxed{h^{-1}(x,y)=\log\sqrt{x^2+y^2},}
$$
the restriction of a continuous function on the punctured plane. Hence $h$ is a <homeomorphism> and \b[the spiral is homeomorphic to the <real line>]. Its accumulation at the origin causes no failure of this argument: the origin is not included in $X$.
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