Question

Ln(1/e^2) without calculator

Answer 1

Once you remember the definition of a log, the answer to this question will literally fall out of your pencil.

First, ' Ln ' means 'natural log' ... logs to the base of ' e '.

Definition of the natural log of a number:
In order to get the number, what power do I have to raise ' e ' to ?

OK.  What power do you have to raise ' e ' to in order to get  1/e²  ?

Isn't  1/e²  the same thing as  e⁻²  ?

So, in order to get  1/e² , you have to raise ' e ' to the  -2  power .

In math-speak:    Ln(1/e²) = -2 .

Answer 2

Ln 1/e²
             =  Ln 1 -  Ln e²       as  Ln X/Y = Ln X  -  Ln Y
             =  0  -  2 Ln e         as Ln 1 = 0    and  Ln X²  = 2 Ln X
                = - 2  * 1          as Ln e  = 1

        = -2