logarithm

log_x[ x^1/2 / x ]  =  a     --->     x^1/2 / x  =  x^a 

Since x must be positive:  x^1/2 / x  =  x^-1/2  =  x^a     --->     a  =  -1/2

                                                                                --->    log_x[ x^1/2 / x ]  =  -1/2 

so log means what power of the number under it is the number beside it and we get that sqrt(x) / (x)=sqrt[-2](x) and sqrt[-2](x)=x^-1/2 so log_x sqrt(x) / (x) = x^-1/2