# Conversion formula between attenuation coefficient (cm^-1) and (dBcm^-1MHz^-1)

So, I was presented in class the fact that (at least in Ultrasound), attenuation coefficients $$\mu$$ can be expressed in units $$\text{dBcm}^{-1}\text{MHz}^{-1}$$ as opposed to the form I was used to; simply $$\text{cm}^{-1}$$.

For a sanity check, I decided I wanted to work out a formula to convert from one to the other.

For sake of simplicity, let's denote the $$\text{dB}$$ form as $$\mu_{dB}$$ and the original (in $$\text{cm}^{-1}$$) as $$\mu$$.

I know for a fact that:

$$\frac{I(x)}{I_0} = e^{-\mu x}$$

and also (provided, though I wouldn't mind knowing where/how it was derived?)

$$\frac{I(x)}{I_0} \text{[dB]} = fx\mu_{dB}$$ Where $$f$$ is frequency

If I'm not mistaken, the conversion from the original units to $$\text{dB}$$ is:

$$10\text{log}_{10}\big(\frac{I(x)}{I_0}\big)$$

So, substituting in, and equating, I ended up with something looking like this:

$$10\text{log}_{10}\big(e^{-\mu x}\big)=fx\mu_{dB}$$

Which, after rearranging, and changing log base, I end up with

$$\mu_{dB} = -\frac{10}{f\text{ln}(10)} \mu$$

The issue I have here, is that this minus sign has appeared, which (I don't think) should be correct?- Can the Attenuation coefficient be negative?

I feel like I'm missing something stupidly obvious (i.e. can I do this in the first place, or is it just its own thing?), but I just can't see it... Any help here as well would be appreciated! Cheers, all!