Is $y=e^{-x}$ same as $\ln(y)=-x$?

Question

I have $y = e^{-x}$

Is it possible to log both sides and get $\ln(y) = -x$.

I suspect this had led me to a mistake in an exam question. Many thanks

Answer 1

If $A=B $, the only condition you need to log both sides is

$A>0$ or $B>0$.

in your case, $B=e^{-x} $ is always $>0$ thus

we get $$\ln (y)=\ln (e^{-x})=-x $$.

Answer 2

Yes absolutely , you can prove this statement and the proof goes like this

y=e^-x , taking natural logarithm of both sides we get lny=(-x)lne=(-x).