Log vs Decibel Scale

🎯 Learning Objectives:

In this topic, we will look at different ways we can represent the pressure values on a one dimensional scale. We will quickly find out that plotting a vast range of values on a linear scale can lead to loss of resolution at the lower range of values, and we will also see how a logarithmic scale is better suited to represent exponential data. We will learn the basics of logarithms or log, as a substitute for the exponent or power function. We go further by representing pressures not on their own, but as ratios, with the baseline pressure value as the threshold of human hearing. We will further explore a unit that is most commonly associated with measurement and representation of pressure, the bel and the decibel. We will derive the equation of converting pressure into dBSPL, and answer a couple of commonly asked questions in the process.

📘 In the previous topic we had discussed sound pressure on its general form with its SI unit of Pascal and we had listed pressures ranging from the threshold of hearing into the threshold of pain and a few common ones in between.
Let us watch The video lecture to better understand the linear vs logarithmic scale.

We can go further and what we can do now is instead of talking about the absolute pressure value of some source which is not particularly useful. We an instead talk about how much more the pressure value is of that source when compared to a baselines. We need ratios. So let talk about it.

Pressure Ratios:

We warned our scale to start from zero, because it is nice that way, and we need to set the baseline value against which all of the comparisons are made. What value we shall be assigned to the baseline? Perhaps the threshold of human hearing.
Alright let us stick with that. So in our scale any pressure value we want to represent will be compared to the threshold of human hearing at 20 micro Pa, as a ratio like so. To constrain the size of the scale we will take the log of this ratio as,

with $P_o=20 μPa$

Now we have constrained all the pressure values on the table, in sound pressure topic, our new scale between 0 to around 7 and our new table looks like this now.

It turns out, something similar already exists. It is a BEL and,
⭐ It is the relative unit of measurement used to express the ratio of one value of a power or a field quantity to another on a logarithmic scale.

Bell is in fact a unit first used in the measurement of electrical power and telephone applications. So it is immediately apparent how it can be applied to the measurement of sound pressure. If we look closer at the definition above, it says power or field quantity. Pressure is, in fact, a field quantity. Pressure exists in a medium and can be measured at any cross sectional area of the medium. And it is dependent on the position from the source meanwhile the source can be considered to be releasing energy over time.
This is illustrated in the Figure below.

Let us take an example to better understand. Consider this analogy.
A heater in a room warms up the room. The heater is rated to some electrical power level whereby it converts electrical energy into heat over time. This causes the temperature in the room to go up. We can view this from the lens of cause and effect. The heater or the electrical power is the cause, and the change in temperature in the room is the effect. They are completely different units of measurement. Electric power is measured in joules per second or in watts and temperature is measured in Celsius. But there is a definite relationship between them, although the relationship can be complex. The temperature that you feel depends on how well the room is insulated, where the heater is positioned and how far you are from the heater. This exact analogy can be applied to let us say a loudspeaker. The acoustical power of a loudspeaker can be considered the source, measured in watts and the pressure difference that it creates is the effect measured in Pascal on a complex relationship exists between Sound Power and Sound Pressure which depends on how reflective the room is, where the loudspeaker is positioned, and how far you are from it. In short, we can see that the power is actually directly proportional to the square of the pressure (which we will discuss in topic, "Sound Power").

Let usu se this relationship in calculating the Bell.
One Bell is said to be the log of the ratio between two power quantities. If we substitute power with the relationship expressing pressure we get the following

As mentioned before we can substitute the baseline pressure value to the threshold of human hearing at 20 micro Pa. Most of the time we want a little more granularity on a slightly larger range of values so we use the decibel scale instead where decibel is just 1/10 of a bell and can be expressed like this

This is the standard and accepted way of measuring sound pressure.
We can recalculate the values of the entire table above and this is what we get where the decibel of sound pressure levels range from zero decibels at the threshold of human hearing to around 130 decibels at the threshold of pain.

Here is an interesting bit which makes decibels a hairy unit to work with and is a source of confusion for a lot of people out there.
When we are talking about ratio of the pressure values we are dividing one unit by the same unit, which cancels the entire unit out. And thus the decibel is a dimensionless unit. Once it is calculated we would have no way to know what it is a measure off. It cloud very well be a measure of the sound power levels or the sound intensity or even voltage or electrical power as shown in the Figure below. All of which are valid ways of being expressed in decibels.

For instance saying something like "a sound is 60 dB loud" makes no sense at all. Because it does not give us any context of what was initially being measured. To make it crystal clear that we are actually talking about pressure and that we have taken the reference pressure value as 20 micro Pa, we append SPL or sound pressure level after dB. So stating something like "a sound is 60 dBSPL" makes perfect sense since there is no ambiguity about, what we are talking about.
This completes our topic on sound pressure level but before we wrap up, just want to explore a couple of commonly encountered questions.
⭐ Can a decibel pressure level being negative? Absolutely, YES
We have set the baseline or the reference value of 20 micro Pa, the lowest our ears can hear. But our measuring instruments can be tuned to pick up pressure values far lower than this in an anechoic chamber. If the the ratio between the pressure values dips below 1 the log value turns negative.

⭐ What is the change in decibels when the pressure is doubled? If we double the pressure what would that mean on the decibel scale?
Let us find out if a pressure value is twice the original then the calculation just boils down as below

So every time pressure is doubled in Pa the decibel scale increases correspondingly by 6 dB. Similarly, if the pressure is halved the decibel scale reduces by 6 dB.

Here is a question for you.
❓ Does doubling the sound pressure double the perceived loudness?
Well, it is subjective as we have already pointed out before, loudness depends on a lot of factors like the frequency content of the sound. But a pressure increase anywhere between 6 to 10 dB would relate to a perceived doubling of the loads.

🧪 Interactive Demo

🧪 Interactive Demo

📝 Key Takeaways

• Representing sound pressures on a linear scale leads to poor resolution for low values, so a logarithmic scale is preferred for representing the wide dynamic range of sound.
• Instead of using absolute pressure values, it’s often more meaningful to use pressure ratios, comparing a measured pressure ( P ) to a baseline reference pressure ( P_0 = 20 , \mu Pa ) (threshold of hearing).
• Logarithmic representation compresses the scale, making it easier to visualize pressure differences across several orders of magnitude.
• The Bel is a logarithmic unit originally used in electrical measurements; it expresses ratios of power or field quantities (e.g., sound pressure).
• Sound pressure is a field quantity, and sound power is proportional to the square of pressure ( P \propto p^2 ).
• Decibels (dB) are 1/10 of a Bel and are the standard for expressing sound pressure levels:

• Decibels are dimensionless, since they are derived from a ratio of the same physical units. For clarity, we add SPL (Sound Pressure Level) to specify the context.
• Negative decibel values occur when measured pressures are below the reference pressure (e.g., in anechoic chambers).
• Doubling the pressure corresponds to an increase of approximately +6 dB; halving corresponds to −6 dB.
• Doubling sound pressure does not necessarily double perceived loudness—loudness perception depends on factors like frequency and human hearing sensitivity.

🧠 Quick Quiz

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