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I want to plot the labels and the graduations of an historical timeline onto a logarithmic spiral.

If this timeline is on the $x$-axis, $-\infty$ would project to the center of the spiral, $+\infty$ to the "point" of infinite radius, and, ideally, the origin to a point whose I could set the radius as a parameter. Morevover, the reverse transformation should map any point of the plane to the band between the $x$-axis and the line $y=b$, where $b$ is a given constant.

What transformation is required?

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