Question

How do you condense `ln x + ln (x-2) - 5 ln y`?

Answer 1

Use a few properties of logs to condense `lnx+ln(x-2)-5lny` into `ln((x^2-2x)/(y^5))`.

Explanation:

Begin by using the property `lna+lnb=lnab` on the first two logs:
`lnx+ln(x-2)=ln(x(x-2))=ln(x^2-2x)`

Now use the property `alnb=lnb^a` on the last log:
`5lny=lny^5`

Now we have:
`ln(x^2-2x)-lny^5`

Finish by combining these two using the property `lna-lnb=ln(a/b)`:
`ln(x^2-2x)-lny^5=ln((x^2-2x)/(y^5))`