Question

How do you condense `3 [ln(x) - 2ln(x^2 + 1)] + 2ln(5)`?

Answer 1

`ln((25x^3)/(x^2+1)^6)`

Explanation:

We have,

`color(red)((1) .lnX^n=nlnX`

`color(red)((2).ln(X*Y)=lnX+lnY`

`color(red)((3).ln(X/Y)=lnX-lnY`
Now,
`3[lnx-2ln(x^2+1)]+2ln(5)`
`=3[lnx-ln(x^2+1)^2]+ln(5)^2...to`,Apply(1)
`=3[ln(x/((x^2+1)^2))]+ln(25)....to` Apply(3)
`=ln((x/(x^2+1)^2)^3)+ln25`
`=ln(x^3/(x^2+1)^6)+ln25`
`=ln(25*x^3/(x^2+1)^6)...to` Apply(2)
`=ln((25x^3)/(x^2+1)^6)`