broggesy9

2022-04-28

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

The product rule for logarithms states that:

${\mathrm{log}}_{b}M+{\mathrm{log}}_{b}N={\mathrm{log}}_{b}(M\cdot N)$

Most sources that I've found* state that this rule only applies when M and N are positive. It's true that${\mathrm{log}}_{b}0$ is undefined, but negative values in place of M and N seem to work just fine:

$\mathrm{log}(-1)+\mathrm{log}\left(4\right)=\mathrm{log}(-1\cdot 4)$

Why the discrepancy?

The product rule for logarithms states that:

Most sources that I've found* state that this rule only applies when M and N are positive. It's true that

Why the discrepancy?

Draidayerabauu

Beginner2022-04-29Added 9 answers

Try this in Wolframalpha.

$\mathrm{log}(-1)+\mathrm{log}(-2)=\mathrm{log}((-1)\times (-2))$

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$z\in \mathbb{C}$ can also be written as $z\cdot {e}^{2k\pi i}$ , where $k\in \mathbb{Z}$ . This results in a multi-valued logarithm function. For instance,

$\mathrm{log}\left(2\right)=\mathrm{ln}\left(2\right)+2k\pi i$

where$k\in \mathbb{Z}$ and $\mathrm{ln}\left(2\right)$ denotes the real value $\approx 0.69$

However, if we restrict our domain to positive reals, and say we are only interested in real valued logarithm, then we have

$\mathrm{ln}\left(ab\right)=\mathrm{ln}\left(a\right)+\mathrm{ln}\left(b\right)$

where$a,b\in {\mathbb{R}}^{+}$ It is also important to note that the imaginary part of $\mathrm{log}\left(z\right)$ is an integer multiple of $2\pi$ if and only if z is a positive real number.

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

where

However, if we restrict our domain to positive reals, and say we are only interested in real valued logarithm, then we have

where

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,

$y={\mathrm{log}}_{b}x{\textstyle \phantom{\rule{2em}{0ex}}}\text{means}{\textstyle \phantom{\rule{2em}{0ex}}}{b}^{y}=x.$

Since${b}^{y}>0$ for all $y\in \mathbb{R}$ as a function of a real variable, ${\mathrm{log}}_{b}x$ is only defined for $x>0.$

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,

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