The Logarithm of Subtraction of two variables
What is the result of this expression , It should mention that the logis 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?
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$\begingroup$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.
$\endgroup$ $\begingroup$We can't really make it any simpler, sadly.
Also, notationally, we might write $e^{-x}$ or $\exp(-x)$, but not $\exp^{-x}$.
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