TITLE: What operations on an equation cause it to be destroyed?
QUESTION [6 upvotes]: I approached my calculus professor about something he said which didn't make much sense to me - He says that in the process of calculating $\lim_{x\to\infty} f(x)^{g(x)}$, you can convert it to $\lim_{x\to\infty}e^{g(x)\cdot \ln(f(x))}$. I understand that much - $e$ and $\ln$ are inverse functions, so they cancel each other out and the end result is the same equation.But when he was showing us how to do implicit differentiation, he says that $y=\ln(f(x))$ can be re-written as $e^y=e^{\ln(f(x))}$, or ultimately $e^y=f(x)$. Same goes for converting a function $y=f(x)^{g(x)}$ to $\ln(y)=g(x)\cdot \ln(f(x))$.
My first impression was to say WTF? I would have thought that raising both sides of the equation to a power of $e$ would destroy the equation... but apparently not. So my question is, what operations can you safely do like this to both sides of an equation - or if it's more concise, what list of things can you NOT do? The only two I'm aware of currently after speaking with my prof are squaring and taking the square root (or cubing, etc..)

REPLY [2 votes]: One operation that will destroy an equation is dividing both sides of the equation by zero! (It makes no sense to equate undefined with undefined!)
Seriously, sticking with defined operations and manipulations on expressions: 
Manipulating each side of the equation in the identical manner maintains "an" equality, but not necessarily the equality.
Manipulations sometime fail to preserve all the information of the original equation (like all potential solutions). 
And at other times such manipulations, in fact, introduce solutions to the manipulated equation that are not solutions to the original equation.

E.g., If $x = y$, then $x^2 = y^2$: this is a true statement. However $(-2, 2)$ solves $x^2 = y^2$, but is not a solution to $x = y$.