Alugaritm agaman

## ln x ≈ log₄ x + log₃₆ x
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 : \ln(). 

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

## Amzruy
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.

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

\ln a = \int_1^a \frac{1}{x}\,dx.
 

Iɣ darnɣ a rad nini mas d anrar lli darnɣ illan iga uzdir.

Tṭṭaf tasɣnt talugaritmiyt aydatn kullu tn n tsɣnin tilugaritmin.

\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 x = ta ɣ tfult tiss snat, zun d mk ad:

\begin{align}
\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{align}
 

## Aydatn

#### Taɣlalt d tsbk n tasɣnt n ulugaritm agaman
Zɣ aylli izrin ar nttafa mas d tasɣnt x \mapsto ln(x)  tlla ɣar ɣ uzilal ]0 ; + \infty[   , nzḍar ad tt nzllm ɣ ammas n uzilal ad nit d:

\forall x \in \R^*_+,\ \ln'(x) = \frac 1x 

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

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

\ f(x) = \ln(x)+k \ / \ k \in \ R

Acku f(1) =k  rad nini mas d ln(a)=k  , s yat tɣarast igan tamatayt:

* \forall (a;b)\in ]0 : + \infty[^2,\ \ln(ab) = \ln(a) + \ln(b)

Zɣ uyda yad rad naf aydatn ddaw as:

* \forall (a;b)\in ]0 : + \infty[^2,\ \ln\left(\frac ab\right) = \ln(a) - \ln(b)

Iɣ iga n  amḍan waxiḍ:

* \forall a \in ]0 ; + \infty[,\ \forall n \in \Z,\ \ln(a^n) = n \ln(a)
* \forall a \in ]0 ; + \infty[,\ \forall r \in \Q,\ \ln(a^r) = r \ln(a)

Iɣ iga n  amḍan ana:

* \forall a \in \R^* ,\ \forall \; n \in \Z^*,\ \ln (a^{n}) = 2n \ln |a| 
* \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) 

\prod ad igan tamatart n ufaris.

#### Aydatn yaḍnin
* \ln 1 = 0
* \ln e = 1
* \ln(xy) = \ln x + \ln y \quad \text{i }\; x > 0\;\text{d }\; y > 0
* \ln(x^y) = y \ln x \quad \text{i }\; x > 0
* \ln x 
* \lim_{x \to 0} \frac{\ln(1+x)}{x} = 1
* \lim_{\alpha \to 0} \frac{x^\alpha-1}{\alpha} = \ln x\quad \text{i }\; x > 0
* \frac{x-1}{x} \leq \ln x \leq x-1 \quad\text{i}\quad x > 0
* \ln{( 1+x^\alpha )} \leq \alpha x \quad\text{i}\quad x \ge 0\;\text{d }\; \alpha \ge 1
## Amawal

*Agaman : naturel
*Tasɣnt, Tawwuri : fonction
*Afaris : produit
*Timrnit : somme
*Azadur : base
*Tamnzut : fonction primitive
*Azilal : intervalle
*Amusnak : mathematicien
*Taflwit : tableau
*Tazmilt : notion
*Asnml : definition
*Anrar : aire
*Ayda : propriete
*Aɣrd : integrale
*Tafult : partie
*Amutti : variable
*Taɣlalt : continuite
*Uzdir : negatif
*Tsbk : monotonie
*Zllm : deriver
*Igmmn : croissante
*Umnig : positif
*Tamhlt : opperation
*Amḍan : nombre
*Amatay : general
*Waxiḍ : impair
*Ana : paire


## Tisaɣulin