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Question 159162: Explain the difference between a logarithm of a product and the product of logarithm. Please give examples
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! log of (a*b) = log(a) + log(b)
log(a) * log(b) is simply the product of the 2 logarithms.
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an example of log of (a*b) is:
let a = 5
let b = 7
then log (5*7) = log(35) = 1.544.....
log (5) + log (7) = 1.544.....
they log (5) + log(7) = log (7*5) = log(35)
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an example of the multiplication of 2 logarithms is a property of logarithms where logarithms of any base are proportional to each other.
this means that log(7)/log(5) = ln(7)/ln(5) = log of 7 to the base 3 / log of 5 to the base 3.
i'll use the first 2 because the calculator does those automatically.
log(7)/log(5) = 1.20906.....
ln(7)/ln(5) = 1.20906......
the ratio holds.
if the ratio holds then log(7)*ln(5) = log(5)*ln(7)
1.36013.... = 1.36013.....
they are equal to each other.
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an example where logarithms are divided is the base conversion formula
where log of x to the base b = log of x to the base c / log of b to the base c.
if i want to convert from ln of a number to log of that number to the base 10 i would follow the formula as follows:
let my number = 722.
ln (722) = 6.582025139
if i wanted to convert that to the base 10, i would take log (722) / log(e)
log (722) = 2.858537198
log (e) = .434294482
log (722) / log (e) = 2.858537198/.434294482 = 6.582025139.
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log (a/b) = log (a) - log (b)
log(a)/log(b) is just a division
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i didn't see in the literature where log(a)*log(b) and log(a)/log(b) was used for other than a straight multiplication or division.
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caveat:
from what i researched, i believe this is accurate but i would caution you to confirm with other sources before accepting this as gospel.
i didn't see any other application for log(a)*log(b) or log(a)/log(b) which doesn't mean there isn't any. it just means i didn't see it. it's definitely not one of the basic operations of logarithms because it's not listed as one of them.
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