I've been learning logarithm properties in a finite mathematics course and I am confused about a particular example in the text book. They are using the property $b^{log_b(x)}=x$, $x>0$, to solve the following problem:
$\frac{log_e(x)}{log_e(b)}$
- The first thing they do is replace x in the equation with $b^{log_b(x)}$ giving the result:
$\frac{log_e(b^{log_b(x)})}{log_e(b)}$
- Then use another property ($log_bM^p=p*log_bM$)to get:
$\frac{log_b(x)*log_e(b)}{log_e(b)}$
- The two "$log_e(b)$" cancel out and we are left with the answer
$log_b(x)$
My question is that before #3(or at any point really) I could replace the only "x" in the equation with blogb(x) again could I not? And I could continue replacing "x" with blogb(x) and get a massive equation. What I need to understand is why do they replace x with that logarithm and why can't I continue to replace it over and over(as pointless as that would be). I put in some numbers and constantly changing "x" to that log changes the result so it obviously can't be done continuously. Also in the property it states x>0. Does that mean that after using this property x now is greater than zero. Or can this property only be applied if x is greater than zero. In which case how can I know x=0 if no value has been assigned to x at this point?
I am not great at math and my teachers have told me this textbook has many typos and errors so I end up questioning everything. So any clarification on when I can and cannot use blogb(x)=x would be great thanks.