Sound Intensity and Intensity Level Confusion

[dennis-brodmann]

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At 1 m, the sound intensity is 25 dB for a particular source. The question is asking what the approximate sound intensity is when one is 2 m away from the source.


Although I remember the formulas, the relationship between distance, intensity, and sound intensity level is not coming naturally to me while working on these problems, and when I read the explanations for the mathematical solutions, they don't click.

Here is what I think I know:

I understand that intensity I is inversely proportional to the radius squared (i.e. I is proportional to 1/r^2).
Sound intensity level, B, is equal to 10log(I/Io), where I=intensity of a particular sound and Io=intensity of original sound.

Also, correct me if I am wrong, but
A decrease in intensity by a power of ten, is the same as subtracting the intensity level by the integer multiple of 10.
That is, (I/10^n) = B - n10
An increase in intensity by a power of ten, is the same as adding an integer multiple of ten to the intensity level.
That is, (10^n)(I) = B + n10

For this particular question, the back of my book (TBR by the way), says
that intensity dropped by 1/4. I see that since the radius doubled.
So, if I=new intensity and Io=original intensity, I=(1/4)Io.

But they use the sound intensity level equation to calculate the change in sound intensity level.
Furthermore, they say that this change is equal to 10log(4).
In the sound intensity level equation, the ratio within the logarithm is I over Io...
in this problem, wouldn't that translate to B=10log(1/4)?

 

[LoLCareerGoals]

You can say it dropped by 10*log(4) or it changed by 10*log(1/4) it is just a sign issue.
If they are asking for a magnitude of change (without the sign), then it is 10*log(4).
Just keep in mind that 10*log(4) = -10*log(1/4).

 

[dennis-brodmann]

You can say it dropped by 10*log(4) or it changed by 10*log(1/4) it is just a sign issue.
If they are asking for a magnitude of change (without the sign), then it is 10*log(4).
Just keep in mind that 10*log(4) = -10*log(1/4).

I see that a change in sign results in a flip in the ratio, but how would I automatically know that B = - 10*log(1/4) from the question stem?
Is it because I should know that the further we move from a source, the lower the intensity which implies a subtraction from the sound intensity level?

Also, does B=10log(I/Io) actually represent delta(sound intensity level) since we are taking the ratio of I to Io?