Expand each expression:: log_(9) (6x^(3) y^(5) z^(2)) In(\frac{x^(7) }{\sqrt{&y-1)z^(3) } } ) In(\frac{\sqrt[3]{ab^(2) } }{\sqrt{yz^(5) } } )

Question

Expand each expression::


`log_(9) (6x^(3) y^(5) z^(2))`


`In(\frac{x^(7) }{\sqrt{&y-1)z^(3) } } )`


`In(\frac{\sqrt[3]{ab^(2) } }{\sqrt{yz^(5) } } )`

Answer 1

`log_(9)6+3log_(9)x+5log_(9)y+2log_(9)z`

`7lnx-(1)/(2)ln(y-1)-(3)/(2)lnz`

`(1)/(3)lna+(2)/(3)lnb-(1)/(2)lny-(5)/(2)lnz`

Explanation:

`log_(9)(6x^(3)y^(5)z^(2))=`

`log_(9)6+log_9}x^(3)+log_(9)y^(5)+log_(9)z^(2)=`

`log_(9)6+3log_(9)x+5log_(9)y+2log_(9)z`

`ln\frac{x^(7)}{\sqrt{(y-1)z^(3)}}=lnx^(7)-ln[(y-1)z^(3)]^{(1)/(2)} =`

`7lnx-ln(y-1)^{(1)/(2)}(z^(3))^{(1)/(2)}}`=

`7lnx-ln(y-1)^{(1)/(2)}-lnz^{(3)/(2)}=`

`7lnx-(1)/(2)ln(y-1)-(3)/(2)lnz`

`ln\frac{\sqrt[3]{ab^(2)}}{\sqrt{yz^(5)}}=`

`ln(ab^(2))^{(1)/(3)}-ln(yz^(5))^{(1)/(2)}=`

`ln(a)^{(1)/(3)}(b)^{(2)/(3)}-ln(y)^{(1)/(2)}(z)^{(5)/(2)}=`

`(1)/(3)lna+(2)/(3)lnb-(1)/(2)lny-(5)/(2)lnz`