Applying function to entire side of equation, not just terms

[find_the_fun]

I hate to ask this but whenever applying a function to the equation, the arguments is the entire one side of the equation right?

What I mean is
\(\displaystyle
ln|y|=ln|x|+C\)

can be rewritten as \(\displaystyle e^{ln|y|}=e^{ln|x|+C}\)
but not \(\displaystyle e^{ln|y|}=e^{ln|x|}+e^C\) ?

So the entire RHS or LHS becomes the argument?

Similarly \(\displaystyle \sin{x}=x+y\)
can be rewritten as \(\displaystyle x=\arcsin{(x+y)}\)
but not \(\displaystyle x=\arcsin{x}+\arcsin{y}\)

I keep messing this up and it's really annoying.

 

[Nono713]

Yes, it applies to the entirety of both sides, and the equality will hold for any function applied to both sides. Anything else is not guaranteed to hold for all functions.