Question

How do you solve `lnx + ln(x+1) = ln12`?

Answer 1

`x=3`

Explanation:

In general
`color(white)("XXX")log(A)+log(B) = log(AB)`
(this is one of the basic logarithmic rules)

Specifically
`color(white)("XXX")ln(x)+ln(x+1) = ln(x^2+x)`
and we are told, this is
`color(white)("XXXXXXXXXXXXX")=ln(12)`

`rArr x^2+x = 12`

`color(white)("XXX")x^2+x-12=0`

`color(white)("XXX")(x+4)(x-3)=0`

`color(white)("XXX")x=-4` `color(white)("XXX")`or`color(white)("XXX")x=3`

Since `ln(x)` is not defined for negative values of `x`
`rArr x=3`