Restrictions on laws I'm wondering about the restrictions, my doubt is for example at $\log_a(b)=c\implies a^c=b$, how would anyone add the restrictions for this? I know the argument and the base of a log have to be >0 excluding 1, hence $a,b\in (0,+\infty)\setminus \{1\}$  but wouldn't that mean that $1^1=b$ is not valid? while it is obviously valid for b=1? I don't know i'm just confused, hopefully someone shines a light on everything, and if it's not much to ask, please tell me how to express restrictions on such complex situations? Another example would be $x^y=0$ the restrictions in my head would be $x,y>0$ because we want to avoid cases like $0^0$ or $0^{-2}$ BUT, according to what i wrote x can't be 0, but for example when x=0, $0^y=0$ is true for y>0, but WE SAID that x couldn't be 0! So i'm just in general confused by all this, hopefully the 2 examples i gave, helps to visualise the confusion i'm having better.

The logarithmic function and the exponential function are inverse functions between each other.
If a function is  bijective it has an inverse.
So definition of exponential function must have some restriction to make this function bijective.
$y=a^x$, $x$ a real number(domain), ($a>0$ but $a\not=1$) $y=a^x>0$(range) is a bijection
And its inverse $x=log_ay$   so $y>0$(domain), $x$ is a real number (range) and (($a>0$ but $a\not=1$)) same as exponential function
