Q-210956-im-read-leithold-book-again-in-this-book-1-define-ln-x-1-x-dx-x-x-gt-0-2-define-ln-e-1-1-e-dx-x-3-define-exp-x-y-ln-y-x-d-ln-u-du-1-u-d-ln-u-dx

Question Number 210967 by mahdipoor last updated on 24/Aug/24

Q . 210956

i m r e a d l e i t h o l d b o o k a g a i n, i n t h i s b o o k :

1} d e f i n e : l n (x) = \int_{1}^{x} d x / x x > 0

2} d e f i n e : l n (e) = 1 = \int_{1}^{e} d x / x

3} d e f i n e : e x p (x) = y \Leftrightarrow l n (y) = x

\frac{d (l n (u))}{d u} = \frac{1}{u} \Rightarrow \frac{d (l n (u))}{d x} = \frac{d u / d x}{u} \Rightarrow

u = x^{r} \Rightarrow \frac{d (l n (x^{r}))}{d x} = \frac{{r x}^{r - 1}}{x^{r}} = r \times \frac{1}{x} = r \times \frac{d (l n (x))}{d x}

\Rightarrow l n (x^{r}) = r l n (x) + K \Rightarrow x = 1 \Rightarrow K = 0

\Rightarrow l n (x^{r}) = r \times l n (x) \forall x > 0, \forall r

g e t x = e \Rightarrow l n (e^{r}) = r \times l n (e) = r \Rightarrow e x p (r) = e^{r}

\Rightarrow e x p (x) = e^{x} = y \forall x

{l o g}_{e} (e^{x}) = {l o g}_{e} (y) = x a n d d e f i n e : l n (y) = x

\Rightarrow\Rightarrow\Rightarrow l n (y) = {l o g}_{e} (y) = x