MHB Applying function to entire side of equation, not just terms

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When applying a function to an equation, the function must be applied to the entire side of the equation, not just individual terms. For example, in the equation ln|y| = ln|x| + C, it is correct to rewrite it as e^(ln|y|) = e^(ln|x| + C), but incorrect to separate the terms as e^(ln|y|) = e^(ln|x|) + e^C. Similarly, for sin{x} = x + y, it can be rewritten as x = arcsin(x + y), but not as x = arcsin(x) + arcsin(y). This principle ensures that the equality is maintained when applying functions. Misapplying this can lead to incorrect conclusions in mathematical reasoning.
find_the_fun
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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
$$
ln|y|=ln|x|+C$$

can be rewritten as $$e^{ln|y|}=e^{ln|x|+C}$$
but not $$e^{ln|y|}=e^{ln|x|}+e^C$$ ?

So the entire RHS or LHS becomes the argument?

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

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