The Decibel (dB) Scale: A Logarithmic Scale for Sound

The sound level \(L\) in decibels is defined as:

\( L = 10 \log_{10}\left(\dfrac{I}{I_0}\right) \)

where \(I\) is the intensity of the sound being measured and \(I_0\) is a fixed reference intensity (the quietest sound a healthy ear can detect). Because \(I_0\) is constant, the whole formula depends only on the ratio \(\dfrac{I}{I_0}\).

When loudness is measured using amplitude (such as voltage or pressure) instead of intensity, the formula uses a factor of 20 instead of 10:

\( L = 20 \log_{10}\left(\dfrac{A}{A_0}\right) \)

The factor changes from 10 to 20 because intensity is proportional to the square of amplitude, and squaring inside a logarithm doubles the coefficient out front.

Human hearing itself responds to sound roughly logarithmically: a sound that feels "twice as loud" to your ear usually corresponds to roughly 10 times the intensity, not double the intensity. Using a linear scale for intensity would mean writing numbers with a dozen zeros to describe everyday sounds. The decibel scale fixes this by turning multiplication into addition through the logarithm, so every step of 10 dB represents intensity multiplied by 10.

This is the same idea behind converting a logarithm to exponential form: a logarithmic scale takes a wide-ranging exponential relationship and turns it into a simple, evenly spaced scale.

Suppose a sound has an intensity 1000 times the reference intensity \(I_0\). Find the decibel level.

\( L = 10 \log_{10}\left(\dfrac{I}{I_0}\right) = 10 \log_{10}(1000) \)

Since \(1000 = 10^3\), \(\log_{10}(1000) = 3\), so:

\( L = 10 \times 3 = 30 \) dB\( \)

An intensity ratio of 1000 to 1 corresponds to a 30 dB sound level.

A sound is measured at 60 dB. What is its intensity ratio \(\dfrac{I}{I_0}\)?

\( 60 = 10 \log_{10}\left(\dfrac{I}{I_0}\right) \)

Divide both sides by 10:

\( \log_{10}\left(\dfrac{I}{I_0}\right) = 6 \)

Now rewrite the logarithmic equation in exponential form:

\( \dfrac{I}{I_0} = 10^6 \)

A 60 dB sound is one million times more intense than the reference sound \(I_0\). This step-by-step isolation is exactly the technique used when you solve logarithmic equations in general.

Because the decibel scale is logarithmic, small differences in dB represent large differences in actual loudness. The table below gives typical approximate levels.

Notice that going from a whisper (30 dB) to a rock concert (110 dB) is an 80 dB jump, but the intensity ratio has increased by a factor of \(10^{8}\), or 100 million. That gap is what a logarithmic scale is designed to handle compactly.

The decibel scale always compares a measured intensity to a fixed reference intensity, never a raw absolute number. Every 10 dB increase multiplies intensity by 10, every 20 dB increase multiplies intensity by 100, and so on. Working with dB problems is really just evaluating logarithms and solving simple logarithmic equations dressed up in a physics context.

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