Question

If a2+b2=23ab show that:
log a+b÷5=1÷2(log a+log b)

Answer 1

Because the right side has log(a) and log(b), we must have a>0 and b>0 (this is used only to justify taking log of the product ab, and then writing it as log(a) + log(b))

a² + b² = 23ab
a² + 2ab + b² = 25ab
(a + b)² = 25ab
(a + b)² / 25 = ab
(a + b)² / 5² = ab
[(a + b)/5]² = ab

Take log of both sides:

log ([(a + b)/5]²) = log(ab)
2 log((a+b)/5) = log(a) + log(b)
log((a+b)/5) = ½[log(a) + log(b)]