`((log)_2 3)dot((log)_3 4)dot((log)_4 5)(log)_n(n+1)=10.` Find `n=?`

The value of 3^((log)_4 5)+4^((log)_5 3)-5^((log)_4 3)-3^((log)_5 4) is- a.0 b. 1 c.2 d. none of these

Prove that ((log)_(a)N)/((log)_(ab)N)=1+(log)_(a)b

Which of the following when simplified, vanishes? 1/((log)_3 2)+2/((log)_9 4)-3/((log)_(27)8) (log)_2(2/3)+(log)_4(9/4) -(log)_8(log)_4(log)_2 16 (log)_(10)cot1^0+ (log)_(10)cot2^0+(log)_(10)cot3^0++(log)_(10)cot89^0

|(log)_3 512(log)_4 3(log)_3 8(log)_4 9|xx|(log)_2 3(log)_8 3(log)_3 4(log)_3 4|= (a) 7 (b) 10 (c) 13 (d) 17

In n = 10!, then what is the value of the following? (1)/(log_(2)n) +(1)/(log_(3)n) +(1)/(log_(4)n)+…..+ (1)/(log_(10)n)

Show that: (1)/(log_(2)n)+(1)/(log_(3)n)+(1)/(log_(4)n)+...+(1)/(log_(43)n)=(1)/(log_(43!)n)

If n in N, prove that 1/(log_2x)+1/(log_3x)+1/(log_4x)++(1)/(log_n x)=1/(log_(n !)x)

Comprehension 2 In comparison of two numbers, logarithm of smaller number is smaller, if base of the logarithm is greater than one. Logarithm of smaller number is larger, if base of logarithm is in between zero and one. For example log_2 4 is smaller than (log)_2 8 a n d(log)_(1/2)4 is larger than (log)_(1/2)8. Identify the correct order: (log)_2 6 (log)_3 8> log_3 6>(log)_4 6 (log)_3 8>(log)_2 6> log_3 6>(log)_4 6 (log)_2 8<(log)_4 6

The sum of the series: (1)/(log_(2)4)+(1)/(log_(4)4)+(1)/(log_(8)4)+...+(1)/(log_(2n)4) is (n(n+1))/(2) (b) (n(n+1)(2n+1))/(12) (c) (n(n+1))/(4) (d) none of these