# Approximate formula to find velocity from cosmological redshift

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?

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.