Ln (1/e^2) without calculator

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

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.