What is ln(e^{3x}(1-2x)^{-2}) ?

The solution to ln(e^{3x}(1-2x)^{-2}) is 3x-2ln(1-2x)

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simplify ln(e^{3x}(1-2x)^{-2})

simplify

ln (e^{3 x} (1 - 2 x)^{- 2})

3 x - 2 ln (1 - 2 x)

Solution steps

ln (e^{3 x} (1 - 2 x)^{- 2})

Apply log rule:

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

ln (e^{3 x} (1 - 2 x)^{- 2}) = ln (e^{3 x}) + ln ((1 - 2 x)^{- 2})

= ln (e^{3 x}) + ln ((- 2 x + 1)^{- 2})

Apply log rule:

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

ln (e^{3 x}) = 3 x ln (e)

= 3 x ln (e) + ln ((1 - 2 x)^{- 2})

Apply log rule:

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

ln ((1 - 2 x)^{- 2}) = - 2 ln (1 - 2 x)

= 3 x ln (e) - 2 ln (1 - 2 x)

3 x ln (e) = 3 x

= 3 x - 2 ln (- 2 x + 1)

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What is ln(e^{3x}(1-2x)^{-2}) ?
The solution to ln(e^{3x}(1-2x)^{-2}) is 3x-2ln(1-2x)