Solve exponential equation $3^x= 2^x+2$ How do we solve this? I can't think of an easy way.. Is there any way to solve it without using newton's method or other approximations?
$3^x=2^x+2$

First of all, notice that the Newton-Raphson Method will almost never solve an equation. It will give you some very good numerical approximations, but it will almost never find the exact answer.
There is no algebraic way of solving the equation $3^x=2^x+2$. There are, of course, may ways of finding numerical approximations. You could use:


*

*Interval Bisection

*Linear Interpolation
I am unsure as to why you don't want to use the N-R-Method. It is a very good method, especially when solving equations like the one you have given. 
In my humble opinion, the best way to approximate a solution is through the N-R-Method. I get
$$x \approx 1.39495460217075$$
