La función logaritmo natural by anmilnet

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castellano · @anmilnet · (edited)
$4.01
La función logaritmo natural
![La función logaritmo natural.jpg](https://steemitimages.com/DQmZKpPDxbyo1PdsQ1fDeao8MwrXVSp7BNUqScgzyGB9hTH/La%20funci%C3%B3n%20logaritmo%20natural.jpg)

En esta oportunidad quiero presentar el tema del logaritmo natural, para ello tracemos la gráfica de la función f(t)=1/t , donde t ≠0, y enfoquemos nuestra atención en el primer cuadrante del **sistema de ejes coordenados**.
Veamos la imagen:
<center>![Función logaritmo natural.jpg](https://steemitimages.com/DQmYysQurwCpNU38tfc7kdWoJ1sYDZvjY3JDPd7e7Pd49Ue/Funci%C3%B3n%20logaritmo%20natural.jpg)</center>
Centremos nuestra atención en el área **R**  bajo esa curva, limitada por las rectas x=1 y x, donde  x>1 (pero x también puede ser 0<x<1)
En la siguiente imagen se encuentra sombreada el área **R**:
<center>![R.jpg](https://steemitimages.com/DQmZAtrPZNeJDpB5irrhc1YusDUEMMdyydLAHp6bRfqzTN8/R.jpg)</center>
Esa área **R** limitada por las rectas x=1 , x donde x puede ser mayor o menor que 1 (x siempre será mayor que 0) y la curva y= 1/t ( donde t debe ser diferente de 0);recibe el nombre de logaritmo neperiano de x, y se denota **Lnx**.

Es decir que el logaritmo neperiano es un área entre curvas, por ello se expresa como una integral definida, ver fórmula en la imagen.

Si aplicamos derivada en ambos lados de la igualdad mostrada en la imagen:
![Derivando al logaritmo natural.jpg](https://steemitimages.com/DQmYZ86qrG9ACg5tue49GDaLGiBUn7vUMQ32GUVNgcBWcfe/Derivando%20al%20logaritmo%20natural.jpg)
Y aplicando **Regla de la Cadena**, vemos que:
![Derivada del logaritmo natural Regla de la cadena.jpg](https://steemitimages.com/DQmeGJMnnhHRfUupw4Qk7XWBLSTP1oX8RrDabZimrSXhMth/Derivada%20del%20logaritmo%20natural%20Regla%20de%20la%20cadena.jpg)
Veamos un ejemplo de aplicación:
D<sub>x</sub>Ln(5x<sup>4</sup>-1)=[1/(5x<sup>4</sup>-1)]D<sub>x</sub>[(5x<sup>4</sup>-1]=[1/(5x<sup>4</sup>-1)][20x]=20x/(5x<sup>4</sup>-1)

También se puede probar que: 
## D<sub>x</sub>Ln|x|=1/x donde x&ne;0.
Supongamos que x>0, entonces |x|=x, y entonces D<sub>x</sub>Ln|x|= D<sub>x</sub>Lnx= (1/x).(1)=1/x
Y si x<0, entonces D<sub>x</sub>Ln|x|= D<sub>x</sub>Ln(-x)= (-1/x)(-1)= 1/x
Por ello:
&int;1/xdx=Ln|x| + C, donde x&ne;0
Generalizando a través de la "Regla de la cadena"
Se concluye que:
## &int;(1/u)du=Ln|u| + C, donde u&ne;0

Veamos el siguiente ejemplo de aplicación.
&int;[x/(4x<sup>2</sup>+1)]dx
Si hacemos u=4x<sup>2</sup>+1
Entonces du= 8xdx
Dsepejando xdx=du/8
Sustituyendo en la integral, se tiene:
&int;[x/(4x<sup>2</sup>+1)]dx=1/8&int;du/u =1/8&int;[1/u]du= 1/8[ln|u| +c]= 1/8[ln|4x<sup>2</sup>+1| +c]
=(1/8)ln|4x<sup>2</sup>+1| +(1/8)c=(1/8)ln|4x<sup>2</sup>+1| +k, donde la constante de integración  k=1/8c



Referencia:
https://docs.google.com/file/d/0B8wx9bEA31Jzc0tyTVlWOHgycnc/edit
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@speedvoter · 
<p>This post has received a 1.27 % upvote from @speedvoter thanks to: @anmilnet.</p>
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vote details (2)
@julioccorderoc · 
Pensaba que la función logarítmica era la inversa a la exponencial, ¿me equivoco? Sin embargo gracias por la información, me animas a investigar mas sobre el tema, en estos momentos estoy estudiando cálculo integral y quiero comprender bien el tema.

Saludos!
👍  
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@alexandermoreno · (edited)
Estas en lo cierto, la función logaritmo natural también es la inversa de la función exponencial. Lo presentado en este post, digámoslo así, es la visión geométrica de la función logaritmo natural.
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@julioccorderoc · 
Tengo grandes dudas con respecto a eso.

Es decir, la integral es la inversa de la derivada, pero también es el área bajo la curva (el limite de una sumatoria) ¿Qué tiene que ver una cosa con la otra? ¿Por qué se relaciona el limite de un cociente (la derivada) con el limite de una sumatoria?
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@elvys · 
Muy Interesante upvote now
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