Question

How do you solve `2(lnx)^2 + lnx =1`?

Answer 1

Use a dummy variable to put the equation into a more familiar form, then substitute back in to get to `x=e^(1/2) and e^(-1)`

Explanation:

Let's start with the original:

`2(lnx)^2+lnx=1`

In one sense, this is a very complex and complicated equation. But in another sense, we can change the form of the equation to a simple trinomial by substituting a dummy variable, say A, for `lnx`. So we can:

`A=lnx`

therefore:

`2A^2+A=1`

now let's solve for A:

`2A^2+A-1=0`

`(2A-1)(A+1)=0`

So `A=1/2 and -1`

Which means that:

`lnx=1/2 and -1`

`x=e^(1/2) and e^(-1)`