Question

a²+b²=7ab show that log(a+b/2)=1/2loga+1/2logb​

Answer 1

Please write the problem less ambiguously: If

a2 + b2 = 7ab

Prove

log( (a+b)/3 ) = (1/2) (log a + log b)

The a+b/3 threw me off for a bit, so I almost didn't respond.

The first two things to consider is simplifying (log a + log b) and taking care of the (1/2) by multiplying both sides by 2.

Remember two important log rules

log xy = log x + log y

n log x = log xn

Notice that the given includes a2 + b2, as well as terms with ab, so it is also worth remembering that

a2 + 2ab + b2 = (a + b)2

The "creative problem solving" part is to add 2ab to both sides, then things should fall in to place.