Prove that `ln(1+x) < x` for `x > 0.`

Prove that log_(e)(1+x)ltxforxgt0

If ("log"x)/(y - z) = ("log" y)/(z - x) = ("log" z)/(x - y) , then prove that xyz = 1.

We can write log"" x/y = log (x.y^(-1 )) Can you prove that log"" x/y = log x - logy using product and power rules.

If y= 2^((1)/(log_(x)4)) then prove that x=y^(2) .

If x^2 + y^2 = 6xy , prove that 2 log (x + y) = logx + logy + 3 log 2

If x=(log)_(2a)a , y=(log)_(3a)2a ,z=(log)_(4a)3a ,prove that 1+x y z=2y z

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

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" 2 + 16 "log" (16)/(15) + 12 "log " (25)/(24) + 7 "log"(81)/(80) = 1 .