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

Prove that ln(1+x) 0.

Prove that ln(1+x) 0.

Prove that ln(1+x) 0.

Prove that 1/(log_(a/b) x)+1/(log_(b/c) x)+1/(log_(c/a) x)=0

Prove that log_(e )((1+x)/(1-x))=2(x+(x^(3))/(3)+(x^(5))/(5)+…)

Prove that ln(1+x)

If 0ltxlt1 , prove that: log(1+x)+log(1+x^2)+log(1+x^4)+… oo=-log(1-x)

Prove that log_a x+log_(1/a) x=0

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

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