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From IOAA 2013 (Greece) Theory question no. 15, they stated that the approximate formula to find velocity from cosmological redshift is $$v = c*log_e (1+z)$$ and that it is often used by cosmologists. I did a quick google search but found nothing similar to this formula.

So, where did this equation come from and is it really often used by cosmologists?

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This formula is exact if the expansion is linear ($a(t) = H_0 t$) and all peculiar velocities are zero. In that case, the comoving distance to the object is $$\int_{t_\text{then}}^{t_\text{now}} \frac{c\,\mathrm dt}{a(t)} = \frac{c}{H_0} \ln \frac{t_\text{now}}{t_\text{then}} = \frac{c}{H_0} \ln (1{+}z)$$ and the present recessional velocity is $H_0$ times that.

In the real world, the expansion has been not too far from linear, and recessional velocities are not too large, so it's reasonably accurate.

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