Jess Tsui
N. Brewer
MAT 266
Honors Project
Logarithms in Decibels
A decibel is a unit of measurement that can be used for many fields but usually for
electronics, music, engineering and mathematics. It can be written in many abbreviations
but one of which is most common is dB. It is a unit that can be determined using
logarithmic scale that shows a ratio of values. These values can either be represented as a
power quantity of a field quantity.
Using a logarithm to calculate the decibels is useful since it can easily convert large
ratios such as 1,000,000:1 to a more manageable number 6. Since the human ear can hear a
large range, it is convenient to use logarithms. The overall gain in decibels of a system with
multiple components can be calculated by taking the sum of decibel gain for each
component.
Calculation of decibels will use different equations according to what values are
used. If the decibels are calculated using a power quantity, such as intensity or power, then
the equation used is ten times log of ratio of powers.
𝑑𝐵 = 10 log(
𝑃𝑜𝑤𝑒𝑟1
)
𝑃𝑜𝑤𝑒𝑟2
Rarely are measurements in power quantities it but can be easily converted to
power quantities by using equation
(𝐹𝑖𝑒𝑙𝑑 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦)2
𝑅
. However when plugged directly into the
equation above, the result is an equation for field quantities. If quantities such as sound
pressure, current, or voltage, then the equation would be twenty times log of ratio of field
quantities.
(𝐹𝑖𝑒𝑙𝑑 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦)2
𝐹𝑖𝑒𝑙𝑑 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦1
𝑅
1
𝑑𝐵 = 10 log (
) = 20 log(
)
2
(𝐹𝑖𝑒𝑙𝑑 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦)
𝐹𝑖𝑒𝑙𝑑 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦2
𝑅
2
When the ratio of quantities is greater than one, the value of decibels is positive. If it
is less than one then the decibel value is negative. When the ratio of quantities is equal to
one then the resulting value of decibels is zero. The ratio does not increase in direct
proportionality to the increase of decibels. A good estimate to calculate an increase by one
decibel would be to increase the value of the ratio by twenty-six percent however when
you are calculating a decrease in a decibel, the value of quantity ratio decreases by twenty
percent. The relation between power ratios and the resulting amount of decibels is shown
in the figure below.
(Wolfe).
Works Cited:
"Decibel." Wikipedia. Wikimedia Foundation Inc., 24 Apr 2011. Web. 25 Apr 2011.
<http://en.wikipedia.org/wiki/Decibel>.
Locher, Brent. "Decibels." Decibel Tutorial. N.p., 2009. Web. 23 Apr 2011.
<http://www.fourier-series.com/rfconcepts/flash_programs/decibel/decibels.html>.
Wolfe, Joe. "dB: What is a decibel?." Physclips. The Australian Learning and Teaching
Foundation, n.d. Web. 23 Apr 2011.
<http://www.animations.physics.unsw.edu.au/jw/dB.htm>.