Understanding why we multiply log(P/Po) by 10

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Discussion Overview

The discussion revolves around the mathematical reasoning behind multiplying the logarithmic ratio of power levels, log(P/Po), by 10 to express it in decibels (dB). Participants explore the definitions and conversions between bels and decibels, as well as the practical implications of using these units in various contexts.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions the multiplication of log(P/Po) by 10, seeking clarification on its necessity.
  • Another participant suggests that bels are too large a unit for practical use, which justifies the use of decibels instead.
  • A different participant challenges the reasoning by asking why the multiplication by 10 is necessary if bels are already deemed too large.
  • One participant draws an analogy between the use of decibels and microfarads, indicating that it is a matter of convenience in dealing with signal ratios.
  • Another participant proposes an analytical approach using dimensional analysis to explain the conversion from bels to decibels, emphasizing that multiplying by 10 is necessary to convert the units correctly.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and reasoning behind multiplying log(P/Po) by 10. While some agree that decibels are more manageable, others question the logic behind the multiplication itself, indicating that the discussion remains unresolved.

Contextual Notes

Participants reference the conversion between bels and decibels, noting that a decibel is one-tenth of a bel. The discussion includes various assumptions about the practicality of using different units without reaching a consensus on the underlying reasons for the multiplication.

dervast
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Hi i want to ask sth for the db
We have denoted db as log10(P/Po)
I want to ask sth if a^x=theta=>x=log(a)theta
then the db is 10*a^x=10*log(a)theta Is that right?
I want to know why we multiply the log(P/Po) by 10? Why we didnt leave it alone?
 
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dervast said:
Hi i want to ask sth for the db
We have denoted db as log10(P/Po)
I want to ask sth if a^x=theta=>x=log(a)theta
then the db is 10*a^x=10*log(a)theta Is that right?
I want to know why we multiply the log(P/Po) by 10? Why we didnt leave it alone?
Because [tex]log\frac{P}{P_0}[/tex], called Bel is too large a unit to be useful. So we use the deciBel, that is more manageable.
 
If bel is too large to be manageable then why we multiplt by 10?
 
Dervast, it is the same thing as microfarads vs farads. Most of the time we deal in signal ratios that make it more suitable to use decibels instead of bels. Same thing with farads and microfarads. However, if you choose NOT to multiply by 10 and work with bels it is up to you. It will work just fine.
 
dervast said:
If bel is too large to be manageable then why we multiplt by 10?
I think you are looking for an analytical reason.
Try applying dimensional analysis.
log (Pout/Pin) = y (Bel) equ (i)
how to convert to dB? Since a dB is 1/10 of a Bel, then 10db = 1 Bel
To convert equat (i) to db, multiply both sides by (10db/1Bel)
10 log (Pout/Pin) = y (Bel)(10db/1Bel) <------dimensions Bel/Bel = 1 leaving just db
10 log (Pout/Pin) = 10y (db) (ii)
right side of equation (ii) means we multiply the solution y in Bels by 10 to obtain a value in dB.
 

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