I find this rather important to some people of openstudy who claim that "anything log,exponentials,arcsin,arccos,arctan...etc
can be taken on both sides of an inequality". I hereby state the general principle.
Suppose we have an inequality over real x and y: x>y.
The above implies f(x)>f(y) if and only if f is a strictly increasing function in [x,y].
The above implies f(x)=y the above statements holds after striking off the "strictly" word.}
Now we have the common usage "take log on both sides" whenever required in some inequalities. Observe that f(x)=log(x) is
an increasing function in [0,+infty). (Remember you can't "take log on both sides" when there is a negative quantity in atleast
one side---see the domain of function definition.)
However "take exponential on both sides" is always allowed since the function f(x)=e^x is increasing everywhere {i.e. in
(-infty,+infty)}. Even the smallest things you apply to inequalities, like adding one on both sides or subtracting 1 on both sides
are possible because the functions f(x)=x+1 and g(x)=x-1 are everywhere increasing.
"Taking square roots on both sides" is permissible when both sides have non-negative numbers. Also the sign of the original
inequality is preserved since f(x)=sqrt(x) is increasing in [0,+infty).
While "taking trigonometric functions" be cautious about the intervals where the function is increasing or decreasing. For x,y
in [-pi/2,pi/2] x>y implies sinx>siny and for x,y in [pi/2,3pi/2] x>y implies sinxy implies x>y-1 {x,y are reals}. You also need precautions when doing this i.e. x>y implies f(x)>y only when f(x) has an
maps x to a "greater" number say x1 so that x1>x>y i.e. f(z)>z for all z in neighbourhood of x {observe that this operation
dilates the inequality; using an operation to make an inequality more stronger than given requires handling with utmost care so
that the inequality does not change its sign!}.
P.S.:I am sorry as this was not a question as you were expecting, but you may give your suggestions. Since this was not a
question, I will give a medal to everyone who comments for about 1 day, after which I will delete this message, unless you
persist. Thank you for reading. Good luck!