Alugaritm agaman

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Iga Alugaritm agaman nɣ Alugaritm anipiri (s tanglizt : Natural logarithm) yat tsɣnt bahra ittyusann ɣ tusnakt. Ar tsnfal tasɣnt ad afaris s tmrnit zun d tisɣnin ilugaritmiyn yaḍnin. Ar tt nttara s mk ad : ${\displaystyle \ln()}$.

Ar nttini is iga alugaritm agaman s uzadur n ${\displaystyle e}$ acku ${\displaystyle ln(e)=1}$. Iga alugaritm agaman tamnzut n tsɣnt : ${\displaystyle x\longmapsto 1/x}$ ɣ uzilal ${\displaystyle ]0;+\infty [}$ .

Alugaritm ad dars yan yism yaḍn ad t igan d "anipiri". Ism ad ikka d John Napir, yan umusnak askatlandi, amskar amzwaru n tflwit talugaritmiyt (tmzaray f talli nssn).

Tettyawskar tazmilt n ulugaritm zɣ dar Grégoire de Saint-Vincent d Alphonse Antonio de Sarasa ɣ usggʷas n 1649.

Ad tili ${\displaystyle a\in ]0;+\infty [}$, nzḍar ad nini mas d asnml n tsɣnt talugaritmiyt iga tt unrar lli illan ɣ izddar n tsɣnt n ${\displaystyle x\mapsto 1/x}$ gr ${\displaystyle a}$ d 1. Ar tt nttara s tɣarast ad:

${\displaystyle \ln a=\int _{1}^{a}{\frac {1}{x}}\,dx.}$

Iɣ darnɣ ${\displaystyle a<1}$ rad nini mas d anrar lli darnɣ illan iga uzdir.

Tṭṭaf tasɣnt talugaritmiyt aydatn kullu tn n tsɣnin tilugaritmin.^[1]

${\textstyle \ln(ab)=\ln a+\ln b.}$

Nzḍar ad nml ayad s tibḍit n uɣrd lli f nsawl ɣ umzwaru s snat tfulin ɣ akud ann nɣrd s usnfl n umutti ${\displaystyle x=ta}$ ɣ tfult tiss snat, zun d mk ad:

${\displaystyle {\begin{aligned}\ln(ab)=\int _{1}^{ab}{\frac {1}{x}}\,dx&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{a}^{ab}{\frac {1}{x}}\,dx\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{at}}\,d(at)\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{t}}\,dt\\[5pt]&=\ln a+\ln b.\end{aligned}}}$

Zɣ aylli izrin ar nttafa mas d tasɣnt ${\displaystyle x\mapsto ln(x)}$ tlla ɣar ɣ uzilal ${\displaystyle ]0;+\infty [}$ , nzḍar ad tt nzllm ɣ ammas n uzilal ad nit d:

${\displaystyle \forall x\in \mathbb {R} _{+}^{*},\ \ln '(x)={\frac {1}{x}}}$

S ɣikAd tilia ${\displaystyle ln}$ tasɣnt tamaɣlalt ɣ uzilal ${\displaystyle ]0;+\infty [}$, acku tazllumt nns tga tumnigt rad nini ma sd ${\displaystyle ln}$ tga tasɣnt igmmn ɣ ammas uzilal ${\displaystyle ]0;+\infty [}$

Ad tg ${\displaystyle f}$ yat tsɣnt ${\displaystyle \ f(x)=\ln(ax)}$ s ${\displaystyle a}$ d ${\displaystyle x}$ sin imḍann umnign. Tazllumt nns tga tazllumt nit n ulugaritm agaman, ilmma:

${\displaystyle \ f(x)=\ln(x)+k\ /\ k\in \ R}$

Acku ${\textstyle f(1)=k}$ rad nini mas d ${\displaystyle ln(a)=k}$ , s yat tɣarast igan tamatayt:

• ${\displaystyle \forall (a;b)\in ]0:+\infty [^{2},\ \ln(ab)=\ln(a)+\ln(b)}$

Zɣ uyda yad rad naf aydatn ddaw as:

• ${\displaystyle \forall (a;b)\in ]0:+\infty [^{2},\ \ln \left({\frac {a}{b}}\right)=\ln(a)-\ln(b)}$

Iɣ iga ${\displaystyle n}$ amḍan waxiḍ:

• ${\displaystyle \forall a\in ]0;+\infty [,\ \forall n\in \mathbb {Z} ,\ \ln(a^{n})=n\ln(a)}$
• ${\displaystyle \forall a\in ]0;+\infty [,\ \forall r\in \mathbb {Q} ,\ \ln(a^{r})=r\ln(a)}$

Iɣ iga ${\displaystyle n}$ amḍan ana:

• ${\displaystyle \forall a\in \mathbb {R} ^{*},\ \forall \;n\in \mathbb {Z} ^{*},\ \ln(a^{n})=2n\ln |a|}$
• ${\displaystyle \forall (a_{1},a_{2},....,a_{k})\in ]0:+\infty [^{k},\ \sum _{n=1}^{k}\ln(a_{n})=\ln(\prod _{n=1}^{k}a_{n})}$

${\displaystyle \prod }$ ad igan tamatart n ufaris.

• ${\displaystyle \ln 1=0}$
• ${\displaystyle \ln e=1}$
• ${\displaystyle \ln(xy)=\ln x+\ln y\quad {\text{i }}\;x>0\;{\text{d }}\;y>0}$
• ${\displaystyle \ln(x^{y})=y\ln x\quad {\text{i }}\;x>0}$
• ${\displaystyle \ln x<\ln y\quad {\text{i }}\;0<x<y}$
• ${\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}$
• ${\displaystyle \lim _{\alpha \to 0}{\frac {x^{\alpha }-1}{\alpha }}=\ln x\quad {\text{i }}\;x>0}$
• ${\displaystyle {\frac {x-1}{x}}\leq \ln x\leq x-1\quad {\text{i}}\quad x>0}$
• ${\displaystyle \ln {(1+x^{\alpha })}\leq \alpha x\quad {\text{i}}\quad x\geq 0\;{\text{d }}\;\alpha \geq 1}$

1. ↑ https://www.britannica.com/science/logarithm Encyclopedia Britannica. Tettuyẓra ass n 2020-08-29.

Aggur:Tusnakt/Tin imgradn