Product rule for logarithms works on any non-zero value? The product rule for logarithms states

broggesy9

broggesy9

Answered question

2022-04-28

Product rule for logarithms works on any non-zero value?
The product rule for logarithms states that:
logbM+logbN=logb(MN)
Most sources that I've found* state that this rule only applies when M and N are positive. It's true that logb0 is undefined, but negative values in place of M and N seem to work just fine:
log(1)+log(4)=log(14)
Why the discrepancy?

Answer & Explanation

Draidayerabauu

Draidayerabauu

Beginner2022-04-29Added 9 answers

Try this in Wolframalpha.
log(1)+log(2)=log((1)×(2))
This will give you the result is false. The reason being log in the true sense is a multi-valued function. This is due to the fact that any zC can also be written as ze2kπi , where kZ. This results in a multi-valued logarithm function. For instance,
log(2)=ln(2)+2kπi
where kZ and ln(2) denotes the real value 0.69
However, if we restrict our domain to positive reals, and say we are only interested in real valued logarithm, then we have
ln(ab)=ln(a)+ln(b)
where a,bR+ It is also important to note that the imaginary part of log(z) is an integer multiple of 2π if and only if z is a positive real number.
Alice Harmon

Alice Harmon

Beginner2022-04-30Added 12 answers

Nonsense. Don't trust software blindly.
W|A is using the complex logarithm, which is a multi-valued function (meaning that it can give you a value for the log of any nonzero complex number, but it's not the logarithm, and not all of the properties that you expect will hold without caveats).
By definition,
y=logbxmeansby=x.
Since by>0 for all yR as a function of a real variable, logbx is only defined for x>0.

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