Prove that `log_a x * log_b y = log_b x * log_a y`

Prove that: log_(a)x xx log_(b)y=log_(b)x xx log_(a)y

If x = log_(c) b + log_(b) c, y = log_(a) c + log_(c) a, z = log_(b) a + log_(a) b, then x^(2) + y^(2) + z^(2) - 4 =

Prove that: log_(a)x=log_(b)x xx log_(c)b xx...xx log_(n)m xx log_(a)n

Prove that: (log_(a)(log_(b)a))/(log_(b)(log_(a)b))=-log_(a)b

If y=a^(x^(a^x..oo)) then prove that dy/dx=(y^2 log y )/(x(1-y log x log y))

Assuming that all logarithmic terms are define which of the following statement(s) is/are incorrect? (A)log_b(ysqrtx)=log_b y.(1/2log_b x) , (B) log_b x-log_b y=(log_b x)/(log_b y) , (C)2(log_b x+log_b y)=log_b (x^2y^2) , (D) 4log_b x-log_b y=log(x^4/y^-3)

prove that x^(log y-log z)*y^(log z-log x)*z^(log x-log y)=1

Prove the following identities: (a) (log_(a) n)/(log_(ab) n) = 1+ log_(a) b" "(b) log_(ab) x = (log_(a) x log_(b) x)/(log_(a) x + log_(b) x) .

If a,b,c are in G.P.,prove that: log_(a)x,log_(b)x,log_(c)x are in H.P.