What is ln(-1/2 e^{-2x}) ?

The solution to ln(-1/2 e^{-2x}) is -2x-ln(2)

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simplify ln(-1/2 e^{-2x})

simplify

ln (- \frac{1}{2} e^{- 2 x})

- 2 x - ln (2)

Solution steps

ln (- \frac{1}{2} e^{- 2 x})

Apply log rule:

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

ln (- \frac{1}{2} e^{- 2 x}) = ln (\frac{1}{2}) + ln (e^{- 2 x})

= ln (\frac{1}{2}) + ln (e^{- 2 x})

Apply log rule:

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

ln (e^{- 2 x}) = - 2 x ln (e)

= ln (\frac{1}{2}) - 2 x ln (e)

Apply log rule:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

ln (\frac{1}{2}) = ln (1) - ln (2)

= ln (1) - ln (2) - 2 ln (e) x

ln (1) = 0

2 x ln (e) = 2 x

= 0 - ln (2) - 2 x

0 - ln (2) - 2 x = - ln (2) - 2 x

= - 2 x - ln (2)

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