Question

How do you use the properties of logarithms to expand `lnroot4(x^3(x^2+3))`?

Answer 1

Given: `ln(root4(x^3(x^2+3)))`

The root 4 can be written as the `1/4` power:
`ln((x^3(x^2+3))^(1/4))`

The property `ln(a^c) = cln(a)` tells us that the `1/4` comes outside as multiplication:

`1/4ln(x^3(x^2+3))`

Use the property `ln(uv) = ln(u) + ln(v)` to make the two factors within the argument become the sum of two logarithms but both are still multiplied by `1/4`:

`1/4ln(x^3)+1/4ln(x^2+3)`

Use the property `ln(a^c) = cln(a)` to bring the cube outside as multiplication by 3:

`3/4ln(x)+1/4ln(x^2+3)`