Answer:
3.5 log(x) - 5 log(y)
Step-by-step explanation:
Given expression is
[tex]log\left(\sqrt{x^7y^{-10}}\right)[/tex]
The term under the square root can be expanded as follows
[tex]\sqrt{x^7y^{-10}} = \sqrt{x^7} \cdot \sqrt{y^{-10}[/tex]
We have the relationship
[tex]\sqrt{x} = x^{1/2[/tex]
So we get the above expression as
[tex]\sqrt{x^7y^{-10}} = \sqrt{x^7} \cdot \sqrt{y^{-10}}\\\\= x^{7/2} \cdot y^{-5}[/tex]
We have to find [tex]log( x^{7/2} \cdot y^{-5})[/tex]
Using the log rules
log(xᵃ · xᵇ) = log(xᵃ) + log(yᵇ)
log(xᵃ) = alog(x)
log(yᵇ) = blog(y)
we get the result as (7/2)log(x) - 5log(y)
= 3.5 log(x) - 5 log(y)
