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

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

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

simplify

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

2 ln (x) - 2 x

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

= 2 ln (x) + 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)

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

2 x ln (e) = 2 x

= 2 ln (x) - 2 x

lo g_{10} (2) + lo g_{10} (5) - lo g_{10} (3)

lo g_{4} (3 x y)

2 lo g_{a} (7) + lo g_{a} (4) - lo g_{a} (10)

8 lo g_{10} (x) - 7 lo g_{10} (x + 1)

lo g_{4} (10) + lo g_{4} (7) + 4 lo g_{4} (3)