Exponential Proof (2 Viewers)

[Lukybear]

Hey Guys I was just going through some questions and in the solutions i had found this;









The problem starts here for me.




Thus

(note: e^lna = a)

Can someone please explain the last two steps for me? How do we know when to take the base of e. Also which topic are we taught e^lna = a?

 

[daryl-d]

all calculus assumes the base of e, and it is assumed that u know that e to the power of x and lnx are mutually inverse functions..ie they undo each other


hope this helps

 

[annabackwards]

e^ln(a) = a

So if it's e^ln(x^e) = x^e

 

[Drongoski]

Why: e^{ln a} = a ??


One way of looking at this is to realise that the e ^{[ ]} and the ln[ ] functions are mutual inverses of each other.

Thus if we apply ln to a = 13 say, we get ln 13; if we now apply the inverse of the ln[ ] function, we "edit undo" what ln does to the 13, the e^{[ ]} undoes it so that we get back the 13. It is a bit like if we apply the sqrt function to 16, we get 4; when we apply the squaring function, the "inverse" of the sqrt function' to this result 4 we get back 4² = 16.

In the same way: ln ( e^1.7 ) we get back 1.7

So: e^{ln a} = a

Basically: f(f^-1(x) ) = x

............: f^-1(f(x)) = x

 

[Lukybear]

It was never taught to me in the 2 unit and 3 unit course. Is it in any books or is this pre req knowledge? Is it in fitzpatrick?

 

[Drongoski]

It was never taught to me in the 2 unit and 3 unit course. Is it in any books or is this pre req knowledge? Is it in fitzpatrick?

A further example

How do you differentiate y = 3^x ?

1st convert 3^x into e^{[ ? ]} = e^{ln 3x} = e^{x ln 3}

.: d/dx (3^x ) = d/dx (e^{x ln3} ) = ln3 e ^{x ln3} = ln 3 . 3^x

I suppose you shld know this for mx1. Don't know where/which text covers this.

 

[boxhunter91]

x = loga^N .: N=a^X it is assumed that youu know eln^A = A