Okay, being completely honest, i don't know how more to make it clearer, this question has been deleted 3 times, maybe people don't actually read what i say at the beginning, which said perfectly what i wanted. Anyways, for the fourth time, i want to ADD RESTRICTIONS TO THIS LIST OF LAWS, they are all different laws, and if you don't ever know what a restriction is, it means i want to add for which cases the laws listed ACTUALLY WORK, because for example log(-2) does not work. I actually found a nice answer before which suggested this, i made a little change.
For all $x\in (0,\infty)$ $$\log(x^n)=n\cdot \log(x)$$ For all $x, y\in (0,\infty)$ $$\log(x\cdot y)=\log(x)+\log(y)$$ $$\log(\frac{x}{y})=\log(x)-\log(y)$$ For all $x\in (0,\infty)$. For all $b\in (-\infty, \infty)$, for all $c\in (0,\infty)\setminus \{1\}$ $$\log_c(x)=b\implies x=c^b$$ For all $x,a \in (0,\infty)$ For all $b,c\in (0,\infty)\setminus\{1\}$ $$\log_b(a)=\frac{\log_c(a)}{\log_c(b)}$$ $$\log_c(b)=\frac{1}{\log_b(c)}$$ $$\log_e(x)=\ln(x)$$ $$e^{\ln(x)}=x$$ So just for confirming, is everything what i wrote above correct?
That looks good to me. (And I suppose your previous questions were closed because there's not much to do other than link to a page of identities, like here: https://mathworld.wolfram.com/Logarithm.html)
One thing I might do is generalize your (currently) last identity
$$\huge b^{\log_{b}\left(x\right)}=x$$