Logarithmic scale

For a start, I'm going to replace your function letter y with the letter g, because otherwise, I don't know how to refer to the vertical axis! Thus, in what follows, my "y" is the common or garden vertical y, and "g" is the function you're looking for.

I assume you mean that you want the following:
- The x-axis is going to be calibrated in such a way that, in base 10 for example, 100 is twice as far from the origin as 10.
- The y-axis is going to be calibrated in an ordinary linear way
- You want the graph y=f(g(x)) plotted against x on this diagram to look like the graph y=f(x) plotted against x on a regular diagram.

[I'm going to ignore your assertion that you want g(T/2) to be halfway along the logarithmic axis, because that's not consistent with your initial assertion of what you want.]

If I've correctly described what you mean, then what you need is just g(x)=log(x), where the log is taken to the same base as you're going to use for scaling the x-axis.

Imagine lining up your two x-axes one above the other (an ordinary one and a logarithmic one), and say you're taking logs to base 10, and say T=4:

0

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1

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2

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3

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4

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1

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10

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100

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1000

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10000---

You want y=f(x) plotted using the top axis to be identical to y=f(g(x)) plotted using the bottom one.

i.e., you want f(4) to be the same as f(g(10000)), and f(2) to be the same as f(g(100)), etc. Clearly, then, the function g=log(x) (to base 10) does the trick. Because then f(g(10000))=f(log(10000))=f(4), and so on.

Hope that's clear.

By the way, this strikes me as a rather bizarre thing that you're trying to achieve!!!