How do you prove a^{x}=e^{xlna}?

[Lipi]

It's not a homework, I just want to understand the e. Just don't know enough yet to grasp it.
Can someone help me prove that this is eaqual:

a$$^{x}=e^{xlna}$$

for example

10$$^{x}=e^{xln10}$$

How do you come from 10$$^{x}$$ to $$e^{xln10}$$

Thanx.

 

[dodo]

Take logarithms of both sides of the equation, and see what you get.

 

[Lipi]

You get zero on both sides, that's all fine. I already know it's the same.

But if you have a function y=a$$^{x}$$,

how do you get to y=e$$^{xlna}$$ being the same thing?

 

[dodo]

You get zero on both sides...

I don't know what you mean. How come, zero? What is the logarithm of a^x ? And of e^(x ln a) ?

 

[Lipi]

Oh i just got it.:shy:

Just found the rule e$$^{lna}=a$$.

Thanx for the kick in the right direction, i appreciate it

 

[HallsofIvy]

Do you understand WHY e^{ln a}= a? Quite simply because "ln x" is DEFINED as the inverse function to e^x! That should have been one of the first things you learned about logarithms.


(And, just to shortcut possible objections, yes, it is quite possible to define $ln(x)= \int_1^x 1/t dt$ and then define y= e^x to be the inverse of THAT function. But still, because they are inverse function e^{ln a}= a and ln(e^a)= a.

 

[slider142]

(And, just to shortcut possible objections, yes, it is quite possible to define $ln(x)= \int_1^x 1/t dt$ and then define y= e^x to be the inverse of THAT function. But still, because they are inverse function e^{ln a}= a and ln(e^a)= a.

Well, to go even further, the only reason to call it a logarithm is because the integral behaves exactly like one, thus the properties of logarithms in general should have been studied before anything about Euler's number. :D