Why Decibels are used to measure Signal to Noise Ratio?
To express a ratio in dB, the ratio must be unit-less, since the logarithm of the ratio must be taken, so I'm not sure I understand why you're puzzled that we use dB.
dB is often used to express unit-less ratios precisely because of the properties of logarithm.
For example, multiplication becomes addition, division becomes subtraction.
Also, since the the signal my be many orders of magnitude greater than the noise, it is more convenient to express the SNR as, say, 50dB rather than 100,000.
I am puzzled because as you said SNR is a unit-less ratio, but at the same time we express it in dB... If the ratio and its logarithm both do not have a unit, what then is the dB? ".
The phrase "the SNR is 50dB" is equivalent to "10 times the log of the ratio of the signal power to noise power equals 50."
The dB is not a dimensionful unit like a unit of length or of time, it is a dimensionless unit.
The number x is a pure number just as the number \$y = 10 \log(x) \$ is though we might say that "y is just x expressed in dB".
Decibels isn't a "unit" in the sense of meter, Netwons, seconds, etc. It is like percent, dozen, parts per million, and the like. Those are all ways of expressing dimensionless numbers. Decibels happens to be a way to express values on a logarithmic scale, but that doesn't change the fact that there is nothing wrong with having various "units" for dimensionless quantities.
Similarly, radians should not have a unit, but are still expressed as rad for clarity.
More specifically, SNR is measured in dB, because dB are convenient for the situation. dBs are convinient for the situation, as the differences of signal and noise can have a large dynamic range, that is, to be small or very large.
So the SNR of 100000V signal with 1V noise is 100000. We take the logarithm of that number and arrive at 10*log(100000) = 50dB. A much nicer number.
Or some such.
Summarizing the discussion in the comments, quantities can be
- unitless
- have units, that have physical significance (e.g. meters)
- or represent units, the do not represent the physical nature of the phenomena, but describe the way we measure it mathematically (e.g. radians, logarithms etc).
The claim has been made that adding quantities, expressed in different units is always meaningless. This is the same as what I have been thought, but might me a simplification for the young learners, just entering the field. IMHO supercat or kriss should ask this topic as a separate (excellent!) question.