4.27 How combine log terms?

To go from \(\ln (A B)\) to \(\ln A + \ln B\) need to use simplify with ln option but add assumptions that one of the terms is positive. Else it will not do it

restart; 
simplify(ln(A*B),ln); # no change 
simplify(ln(A*B),ln) assuming A>0;  # ln(A) + ln(B) 
simplify(ln(A*B),ln) assuming B>0;  # ln(A) + ln(B)
 

To go from \(\ln ( \frac {A}{B})\) to \(\ln A - \ln B\) need to use simplify with ln option but add assumptions that \(B>0\).

restart; 
simplify(ln(A/B),ln);  # no change 
simplify(ln(A/B),ln) assuming A>0;  # ln(A) + ln(1/B) 
simplify(ln(A/B),ln) assuming B>0;  # do this: -ln(B) + ln(A)
 

To go from \(\ln A + \ln B\) to \(\ln (A B)\) need to use combine with assumptions that either \(A\) or \(B\) is positive, else it will not do it.

restart; 
combine( ln(A) + ln(B),ln); # no change 
combine( ln(A) + ln(B),ln) assuming A>0; # ln(A*B) 
combine( ln(A) + ln(B),ln) assuming B>0  # ln(A*B)
 

To go from \(\ln A - \ln B\) to \(\ln \frac {A}{B} \) need to use combine with assumptions that either \(B\) is positive.

restart; 
combine( ln(A) - ln(B),ln);  #no change 
combine( ln(A) - ln(B),ln) assuming A>0;  # -ln(B/A) 
combine( ln(A) - ln(B),ln) assuming B>0   # use this  ln(A/B)