Convert dB value to linear scale

Question

Does anybody know the math required to convert a logarithmic decibel scale to a linear value between 0 and 1, for an automation lane?

In this particular case - the linear value 1 = +15dB, the linear value 0 = -∞dB.

Answer 1

Practical version

• dB → gain-multiplier:

  g = 2^{d / 6}

• gain-multiplier → dB:

  d = 6 · log₂(g)

I find these definitions far more handy than the ones below: changing the amplitude by a factor of two is quite an intuitively relevant change. But, alas, in pre-computer times people couldn't seem to like logarithms of bases other than ten, so...

Official definition

• dB → gain-multiplier:

  g = 10^{d / 20}

• gain-multiplier → dB:

  d = 20 · log₁₀(g)

IMO base-10 is silly, but if you need to do exact calibrations, better use the official version. (Alternatively, use the base-2 version, but replace 6 with the factor 20 · log₁₀2 ≈ 6.020599913; this is then exactly equivalent to the base-10 definition.)

Answer 2

So if we set that 0 dB gain is 1.0 factor, and -∞ db gain is 0.0 factor, it means that (if we are considering voltage gain as in a mixing desk fader) :

gain = 20.0*log10(factor)

therefore :

factor = 10^(gain/20.0)

If, as described in a comment, the 0 dB gain is at 0.65 factor, it means the ref is 0.65.

gain in db = 20*log10(factor/0.65)

factor = 0.65*10^(gain/20.0)