`log(x^5/(y^3sqrtz))`

First, express the radical as exponent.

`= log(x^5/(y^3z^(1/2)))`

Then, apply the quotient property `( log_b(M/N)=log_b M - log_ N)` .

`=logx^5-log^(y^3z^(1/2))`

For the second logarithm, apply the product property `( log_b (M*N) = log_bM + log_bN)` .

`=logx^5-log(y^3+logz^(1/2))`

`=logx^5-logy^3-logz^(1/2)`

And, apply the exponent property `(log_bM^a= alog_bM)` .

`=5logx-3logy-1/2logz`

Hence, `log(x^5/(y^3sqrtz))=5logx-3logy-1/2logz` .

First of all you need to know three properties:

log_(a)(B/C)=log_(a)(B)-log_(a)(C)

log_(a)(B*C)=log_(a)(B)+log_(a)(C) and

log_(a)(B^C)=(C)log_(a)(B)

log((x^5)/((y^3)(sqrt(z))))

Now, convert everything to exponents to make things easier:

log((x^5)/((y^3)(z^(1/2))))

Then, separate the fraction inside the logarithm:

log(x^5)-log((y^3)(z^(1/2)))

Next, separate the multiplication in the second logarithm:

log(x^5)-(log(y^3)+log(z^(1/2)))

Finally, distribute the negative into the parenthesis and use the last property to move the exponents to the front of the logarithm:

5log(x)-3log(y)-(1/2)log(z)

Therefore, log((x^5)/((y^3)(sqrt(z))))=5log(x)-3log(y)-(1/2)log(z)