Prove that `asqrt(log_a b)-bsqrt(log_b a)=0`

Prove that 2^{{sqrt(log_a 4sqrtab + log_b 4sqrtab)-sqrt((log_a)4sqrt(b/a)+log_b 4sqrt(a/b))}sqrt(log_a b))= { 2 , b gea gt1and 2^(log_b a) , 1 ltblta

Prove that 2^(sqrt((log)_a4sqrt(a b)+(log)_b4sqrt(a b))-(log)_a4sqrt(b/a)+(log)_b4sqrt(a/b))dotsqrt((log)_a b)={2ifbgeqa >1 and 2^(log_a(b) if 1 < b < a

Prove that a^x-b^y=0 where x=sqrt((log)_a b), y=sqrt((log)_b a), a >0, b >0 , a , b!=1

Prove that log_a (ab)+log_b (ab)=log_a (ab).log_b (ab)

Prove that: (log_a(log_ba))/(log_b(log_ab))=-log_ab

Prove that a^(x)-b^(y)=0 where x=sqrt(log_(a)(b)&y)=sqrt(log_(b)(a)),a>0,b>0&a,b!=1

prove that a^(x)-b^(y)=0 where x=sqrt(log_(a)b) and y=sqrt(log_(b)a),a>0,b>0 and a,b!=1

Prove that (log_ab x)/(log_a x)=1+log_x b

Prove that : (viii) (log_(a)x)/(log_(ab)x) = 1+log_(a)b .

Prove that log_a x.log_b y=log_b x.log_a y