if `log((a+b)/2)=1/2(loga+logb),` prove that `a=b`

If log((a+2b)/4)=1/2(loga+logb) ,then prove that a^2+4b^2=12ab

If log((a+b)/(6))=(1)/(2)(loga+logb) , then (a)/(b) + (b)/(a) = ______.

If 2log(a+b)=loga+logb+log8 ,then prove that (a+b)^2=8ab

If log(2a-3b)=loga-logb, then a=

If (loga)/(b-c) = (logb)/(c-a) = (logc)/(a-b) , then prove that a^(a)b^(b)c^(c)=1 .

If (a(b+c-a))/(loga) = (b(c+a-b))/(logb) = (c(a+b-c))/(logc) , then prove that b^(c ) . c^(b) = a^(c ).c^(a) = a^(b).b^(a) .

If a, b, c are distinct positive real numbers each different from unity such that (log_b a.log_c a -log_a a) + (log_a b.log_c b-logb_ b) + (log_a c.log_b c - log_c c) = 0, then prove that abc = 1.

If a^2+b^2=23 a b , then prove that log((a+b))/5=1/2(loga+logb)dot

If a^2+b^2=18ab ,then prove that log((a-b)/4)=1/2(loga+logb)

If a^(2)+b^(2)=23ab , then prove that log(a+b)/5=1/2(loga+logb)