Question

How do you express as a single logarithm & simplify `(1/2)log_a *x + 4log_a *y - 3log_a *x`?

Answer 1

`(1/2)log_a(x)+4log_a(y)-3log_a(x)=log_a(x^(-5/2)y^4)`

Explanation:

To simplify this expression, you need to use the following logarithm properties:

`log(a*b)=log(a)+log(b)` (1)
`log(a/b)=log(a)-log(b)` (2)
`log(a^b)=blog(a)` (3)

Using the property (3), you have:

`(1/2)log_a(x)+4log_a(y)-3log_a(x)=log_a(x^(1/2))+log_a(y^4)-log_a(x^3)`

Then, using the properties (1) and (2), you have:

`log_a(x^(1/2))+log_a(y^4)-log_a(x^3)=log_a((x^(1/2)y^4)/x^3)`

Then, you only need to put all the powers of `x`
together:

`log_a((x^(1/2)y^4)/x^3)=log_a(x^(-5/2)y^4)`