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I have a lens system consisting of 7 elements. However, I cannot take the system apart to measure the distance between them or their focal lengths. I do, however, know the effective focal length of the whole system. Is there a way to calculate the magnification of the whole system (for a given image distance)?

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If your system is equivalent to a single lens with focal lenght $f$, then you can proceed like if you had one only lens with that $f$, so $M_L=\frac{s'}{s}$, with $s$ the distance to the object

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