# How do you expand #ln ((sqrt(a)(b^2 +c^2))#?

##### 1 Answer

Sticking with Real logarithms, this expands as:

#ln(sqrt(a)(b^2+c^2)) = 1/2 ln(a) + ln(b^2+c^2)#

#### Explanation:

If

Assuming we're dealing with Real values here and everything is well defined, we must have

Also note that if

Hence:

#ln(sqrt(a)(b^2+c^2))#

#=ln(sqrt(a))+ln(b^2+c^2)#

#=1/2 ln(a) + ln (b^2+c^2)#

If we allow Complex logarithms, then we might try to say something like:

#=1/2 ln(a) + ln (b+c i) + ln (b - c i)#

based on the fact that

For example, if

#0 = ln(1) = ln(b^2+c^2) != ln(b+ci) + ln(b-ci) = ln(-1)+ln(-1) = 2 pi i#

So this Complex identity does not quite work and is messy to fix.