Rewrite the expression using properties of logarithms.

`ln ((x^2root(3)(x-1))/(sqrt(x+1)^3))`

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You can use following laws of logrithms

`ln(x xx y)=ln(x)+ln(y)`

`ln(x ^m)=mlnx`

`ln(x/y)=ln(x)-ln(y)`

So our problem

`ln((x^2root3(x-1))/sqrt(1+x)^3)=ln(x^2root3(x-1))-ln(sqrt(1+x)^3)`

`=ln(x^2)+ln(x-1)^(1/3)-ln((1+x)^3)^(1/2)`

`=2ln(x)+(1/3)ln(x-1)-ln(1+x)^(3/2)`

`=2ln(x)+(1/3)ln(x-1)-(3/2)ln(1+x)`

Ans.

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