Question

Why isn't `ln(c+d) = lnc + lnd` true?

Answer 1

We shall make `lnc+lnd` equal to a number `x`.

`x=lnc+lnd`

By taking exponentials of both sides:
`e^x=e^(lnx+lnd)`
`color(white)(e^x)=e^lnce^lnd`
`color(white)(e^x)=cd`

Now by taking `ln` of both sides.
`ln(e^x)=x`
`x=ln(cd)`

`ln(cd)!=ln(c+d)` unless `c=d/(d-1)`

Proof:

Let's say `ln(cd)=ln(c+d)`, therefore `cd=c+d`

`cd-c=d`

`c(d-1)=d`

`c=d/(d-1)`