What is ln(e^{-w}(e^{-w}-1)) ?

The solution to ln(e^{-w}(e^{-w}-1)) is -w+ln(e^{-w}-1)

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simplify ln(e^{-w}(e^{-w}-1))

simplify

ln (e^{- w} (e^{- w} - 1))

- w + ln (e^{- w} - 1)

Solution steps

ln (e^{- w} (e^{- w} - 1))

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

ln (e^{- w} (e^{- w} - 1)) = ln (e^{- w}) + ln (e^{- w} - 1)

= ln (e^{- w}) + ln (e^{- w} - 1)

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (e^{- w}) = - w ln (e)

= - w ln (e) + ln (e^{- w} - 1)

w ln (e) = w

= - w + ln (e^{- w} - 1)

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What is ln(e^{-w}(e^{-w}-1)) ?
The solution to ln(e^{-w}(e^{-w}-1)) is -w+ln(e^{-w}-1)