(1) ln e^(-2x) = 8
(2) log_5 (625) = x
The rule is a consequence of the fact that the basic exponential and logarithmic functions are inverses of each other. From what I've seen, it would be best at present if you just memorised it. Understanding migt come in the passing of time.
Using the rule:
$\displaystyle \ln e^{-2x} = -2x \, ....$
$\displaystyle \log_{5} 625 = \log_{5} 5^4 = 4 \, ....$
lne^a=a lne
now we may prove lne=1 so that you may get
lne^a =a lne = a
well lne is actually ln_e(e)
let ln_e(e)=t
then e^t=e
comparing both sides we get t=1
so ln_e(e)=lne=1 so
lne^a=a lne =a hence proved
1) lne^(-2x)=8
-2xlne=8
but lne=1 (already proved)
-2x=8
x=-4