Talk:List of logarithmic identities

Peculiar identities

Why are the identities:

$\log _{b}(a+c)=\log _{b}a+\log _{b}(1+b^{\log _{b}c-\log _{b}a})$

$\log _{b}(a-c)=\log _{b}a+\log _{b}(1-b^{\log _{b}c-\log _{b}a})$

... written like that? It seems rather obvious that they would be better written as:

$\log _{b}(a+c)=\log _{b}a+\log _{b}(1+c/a)$

$\log _{b}(a-c)=\log _{b}a+\log _{b}(1-c/a)$

respectively, since $x^{\log _{x}y}=y$ by definition, and $x^{y-z}=x^{y}.x^{-z}=x^{y}/x^{z}$, meaning that $b^{\log _{b}c-\log _{b}a}=b^{log_{b}c}/b^{log_{b}a}=c/a$ ?

Am I missing something? 130.243.139.62 17:48, 12 October 2007 (UTC)

It looks like you're right, but perhaps I'm missing it too. The only thing I can think of is that perhaps this property applies to both negative and positive arguments (real and complex logs), whereas yours would not. This leads into my question. — Preceding unsigned comment added by Eebster the Great (talk • contribs) 01:59, 13 December 2007 (UTC)

He is right, the identities are derived as follows,

$\log _{b}(a+c)=\log _{b}(a(1+c/a))=\log _{b}a+\log _{b}(1+c/a)$

$\log _{b}(a-c)=\log _{b}(a(1-c/a))=\log _{b}a+\log _{b}(1-c/a)$

The only assumption made here is that a is not zero but this is needed anyway to take the log of a so these should be re-written. —Preceding unsigned comment added by 195.112.46.6 (talk) 18:33, 1 March 2009 (UTC)

He is correct, but the way it was originally written has the only occurrences of a and c appearing in a log. This is useful in computations as computing in the log domain can help with numerical stability when the calculations are done on a computer. So his form is neater, but the original would be used often and that is likely why it was written that way. -173.33.199.131 (talk) 07:46, 31 March 2010 (UTC)

Poor sentence

I hate to be pedantic but I think this is a bit of a poor sentence:

"note: to say the limit of a function "equals infinity" is not strictly correct notation, as "infinity" is not a value. What is meant by the limits equations above is simply that the functions increase/decrease without bound."

For a start, there is nothing wrong with saying that a limit equals infinity because it is just notation, it has a very precice meaning in mathematics. As long as you define things well enough, there is no such thing as incorrect notation. Incidentally, the definition of tending to infinity given is also wrong, increasing without bound is not the same as tending to infinity, the sequence, 0,1,0,2,0,3,.... increases without bound, and so does xsin(x) as x tends to infinity but neither of these functions tend to infinity. —Preceding unsigned comment added by 195.112.46.6 (talk) 18:41, 1 March 2009 (UTC)

Or, if we interpret 'increases without bound' to mean monotonic increase, it's false in a different way: 1,0,10,9,100,99,1000,999,... tends to infinity but does not increase without bound. The note is entirely false: saying a limit equals infinity is strictly correct notation, infinity (in this sense) is a value (in the Extended real number line), and the given statement is not what is meant by convergence to infinity. Even if it was correct, this list is not the place for a discussion of the minutiae of limits: that's what the link to limit of a function is for. I've removed the note. Algebraist 22:26, 2 April 2009 (UTC)

Missing log limits

$\lim _{x\to 0}\log {(1+x)}=x$, but always $\log {(1+x)}<x$.

$\lim _{x\to \infty }\log {(1+1/x)}=x$

— Preceding unsigned comment added by Reddwarf2956 (talk • contribs) 00:24, 29 March 2015 (UTC)

I guess you mean $=0$ or the second one? That follows pretty quickly from the upper bound you gave yourself. I added some inequalities so people may deduce this themselves. Thomasda (talk) 16:19, 16 September 2015 (UTC)

Complex logarithm identities

Using "Log" and ln to refer to \ln and log to refer to log seems rather pointless and impractical. "Log" with a capital letter should just be ln. Otherwise it's very confusing.

Definitions

In what follows, a capital first letter is used for the principal value of functions, and the lower case version is used for the multivalued function. The single valued version of definitions and identities is always given first, followed by a separate section for the multiple valued versions.

ln(r) is the standard natural logarithm of the real number r.

Log(z) is the principal value of the complex logarithm function and has imaginary part in the range (−π, π].

$\operatorname {Log} (z)=\ln(|z|)+i\operatorname {Arg} (z)$

$e^{\operatorname {Log} (z)}=z$

$\log(z)=\ln(|z|)+i\arg(z)$

$\log(z)=\operatorname {Log} (z)+2\pi ik$

$e^{\log(z)}=z$

Harmonic number difference

Hello everyone,

I've been working on revising the "Calculus identities" section to include the identity for Harmonic number difference. To avoid cluttering this talk page and to facilitate detailed feedback, I've drafted the proposed changes in my sandbox. Please view the draft here: Harmonic number difference.

I welcome all suggestions and comments to ensure that the content meets Wikipedia guidelines and that it's accurate and clearly explained. I would be grateful if you could share your feedback and thoughts here on this Talk page.

Thank you for taking the time to review and for your valuable insights!

Best regards. Twoxili (talk) 15:58, 25 April 2024 (UTC)