# How do you prove this

1. Nov 20, 2007

### 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.

2. Nov 20, 2007

### dodo

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

3. Nov 20, 2007

### Lipi

You get zero on both sides, that's all fine. I allready 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?

4. Nov 20, 2007

### dodo

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

5. Nov 20, 2007

### 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

6. Nov 21, 2007

### HallsofIvy

Do you understand WHY eln a= a? Quite simply because "ln x" is DEFINED as the inverse function to ex! 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= ex to be the inverse of THAT function. But still, because they are inverse function eln a= a and ln(ea)= a.

7. Nov 21, 2007

### slider142

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