The Logarithm of Subtraction of two variables

Question

What is the result of this expression , It should mention that the log is natural logarithm. $$ \log\left(\exp(-x) - \exp(-y)\right) $$

Could we use the formula which mentioned in wikipedia about logarithmic identities?

$$ \log_{b}(a -c) = \log_b a + \log_b(1- \frac{c}{a}) $$

and does any body know the refrence of the above mentioned formula in wikipedia?

The formula works well, as $$\log(e^{-x} - e^{-y}) = \log\left(e^{-x}(1-e^{x-y})\right) = -x + \log(1-e^{x-y}).$$ — Pragabhava, Nov 08 '12 at 01:34

score 1 · Accepted Answer · answered Nov 08 '12 at 01:49

1

I assume you are familiar with $$\log r+\log s=\log(rs)$$ Now if you replace $r$ with $a$, and $s$ with $1-(c/a)$, you get $$\log a+\log(1-(c/a))=\log(a(1-(c/a)))=\log(a-c)$$ So that's the source of the identity you quote.

answered Nov 08 '12 at 01:49

Gerry Myerson

179,216

score 0 · Answer 2 · answered Nov 08 '12 at 00:18

0

We can't really make it any simpler, sadly.

Also, notationally, we might write $e^{-x}$ or $\exp(-x)$, but not $\exp^{-x}$.

answered Nov 08 '12 at 00:18

Cameron Buie

102,994

Means that there is no formula for logarithm of subtraction? – ben Nov 08 '12 at 00:22
Exactly. Neither for addition. 'What operation is before +?' – Berci Nov 08 '12 at 00:38

The Logarithm of Subtraction of two variables

2 Answers2