What is ln(-1/5 e^{-5x}) ?

The solution to ln(-1/5 e^{-5x}) is -5x-ln(5)

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simplify ln(-1/5 e^{-5x})

simplify

ln (- \frac{1}{5} e^{- 5 x})

- 5 x - ln (5)

Solution steps

ln (- \frac{1}{5} e^{- 5 x})

Apply log rule:

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

ln (- \frac{1}{5} e^{- 5 x}) = ln (\frac{1}{5}) + ln (e^{- 5 x})

= ln (\frac{1}{5}) + ln (e^{- 5 x})

Apply log rule:

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

ln (e^{- 5 x}) = - 5 x ln (e)

= ln (\frac{1}{5}) - 5 x ln (e)

Apply log rule:

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

ln (\frac{1}{5}) = ln (1) - ln (5)

= ln (1) - ln (5) - 5 ln (e) x

ln (1) = 0

5 x ln (e) = 5 x

= 0 - ln (5) - 5 x

0 - ln (5) - 5 x = - ln (5) - 5 x

= - 5 x - ln (5)

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