Prove: `x^(log(b/c) ) x^(log((c)/(a))) x^(log((a)/(b)))=1`

Prove log_(x)b^(x)=x

Prove "log"(a^(2))/(bc) + "log"(b^(2))/(ca) + "log"(c^(2))/(ab) = 0 .

The value of (6a^((log)_e b)((log)_(a^2)b)((log)_(b^2)a))/(e^((log)_e a(log)_e b)) is (a) independent of a (b) independent of b (c) dependent on a (d) dependent on b

Prove that int_(0)^(1)log((x)/(x-1))dx=int_(0)^(1)log((x-1)/(x))dx . Find the value of int_(0)^(1)log((x)/(x-1))dx

(log x)/( (1 + log x)^(2))

Solve (log)_2(3x-2)=(log)_(1/2)x

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) .

The differential coefficient of f((log)_e x) with respect to x , where f(x)=(log)_e x , is (a) x/((log)_e x) (b) 1/x(log)_e x (c) 1/(x(log)_e x) (d) none of these

Prove that ln(1+x) 0.

[The value of "int(sqrt(x^(2)+1){log_(e) (x^(2)+1)-2log_(e)x})/(x^(4))dx" is equal to "],[" (a) "(2)/(3)(1+(1)/(x^(2)))^(3/2)*{log(1+(1)/(x^(2)))-(2)/(3)}+C],[" (b) "-(1)/(3)(1+(1)/(x^(2)))^(3/2)*{log(1+(1)/(x^(2)))-(2)/(3)}+C],[" (c) "(1+(1)/(x^(2)))^(3/2)*{log(1+(1)/(x^(2)))+(2)/(3)}+C ]