Prove that `log(x^n/y^n)+log(y^n/z^n)+log(z^n/x^n)=0`

The value of the determinant |[log_a(x/y), log_a(y/z), log_a(z/x)], [log_b (y/z), log_b (z/x), log_b (x/y)], [log_c (z/x), log_c (x/y), log_c (y/z)]|

Prove that : log _ y^(x) . log _z^(y) . log _a^(z) = log _a ^(x)

If x,y,z are in HP, then show that log(x+z)+log(x+z-2y)=2 log (x-z)

for x,x,z gt 0 Prove that |{:(1,,log_(x)y,,log_(x)z),(log_(y)x,,1,,log_(y)z),(log_(z) x,,log_(z)y,,1):}| =0

Prove that log_n(n+1)>log_(n+1)(n+2) for any natural number n > 1.

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

Assuming that log (mn) = log m + logn prove that log x^(n) = n log x, n in N

The x , y , z are positive real numbers such that (log)_(2x)z=3,(log)_(5y)z=6,a n d(log)_(x y)z=2/3, then the value of (1/(2z)) is ............

The x , y , z are positive real numbers such that (log)_(2x)z=3,(log)_(5y)z=6,a n d(log)_(x y)z=2/3, then the value of (1/(2z)) is ............

If x_n > x_(n-1) > ..........> x_3 > x_1 > 1. then the value of log_(x_1) [log_(x _2) {log_(x_3).........log_(x_n) (x_n)^(x_(r=i))}]