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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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

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

ln (7) + 4 ln (4)

- lo g_{10} (6.3 \cdot 1 0^{- 9})

6 ln (c) + 6 ln (x)

\frac{π}{8} lo g_{10} (2)

- lo g_{10} (5.2 \cdot 1 0^{- 5})