Answer:
![[ln(x* (x+1)^3)/(x-2)]^{(1)/(2)}](https://img.qammunity.org/2021/formulas/mathematics/college/id2hq2yihqqfuod23yyprou0vc59sn95ei.png)
Explanation:
We have given expression
![(1)/(2)[lnx+3ln(x+1)-ln(x-2)]](https://img.qammunity.org/2021/formulas/mathematics/college/hi2hm1w5l4jfh1dibv1sncobj70mi5ilbv.png)
We have to simplify this expression using logarithmic property
According to logarithmic property when two or more log function are added to each other then the functions are multiplied with each other
So
![(1)/(2)[lnx+3ln(x+1)-ln(x-2)]=(1)/(2)[ln(x* (x+1)^3-ln(x-2))]](https://img.qammunity.org/2021/formulas/mathematics/college/x0bfq3jr6eaqevhjea4jchdq3k8i722dqy.png)
Now again using log property
![(1)/(2)[ln(x* (x+1)^3-ln(x-2))]=(1)/(2)[ln(x* (x+1)^3)/(x-2)]](https://img.qammunity.org/2021/formulas/mathematics/college/er3xtfru8gxo76jfls66p9e1jb503lywuh.png)
Now using exponent property of logarithm