What's the result of stacking ND filters?

Question

If you stack two 3-stop neutral density filters, does it reduce the exposure by 6 stops or 9 stops?

Answer

It's six.

Remember, the stops are already logarithmic. That is, a 3-stop reduction (as from a 3-stop ND filter) is a 2³× loss of light — ¹⁄₈ of the light gets through. A one stop filter halves light, since 2¹ is just 2 (→ ¹⁄₂), and two stop filter is 2² (→ ¹⁄₄ the light). When you stack them together, you're adding the exponents, so 2³ stacked with 2³ is 2⁶ — or ¹⁄₆₄th the light. That's the same as thinking "three stops is one over 2³, or ¹⁄₈, and ¹⁄₈ × ¹⁄₈ = ¹⁄₆₄ — which is one over 2⁶".

But, fortunately (and in fact partly why it's done that way), you don't have to remember all this. Just remember that 2³ + 2³ = 2⁶ — or, 3 stops plus 3 stops is 6 stops.

Of course, this is just the math. In the real world, there may be other practical effects, like vignetting in the corners (due to the increased thickness) or color casts — you're adding more layers for the light to go through, and that takes a toll on image quality. See the comments below.

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