What is log_{e}(-2y*e^{4y-x^2-y^2}) ?

The solution to log_{e}(-2y*e^{4y-x^2-y^2}) is ln(2)+ln(y)+4y-x^2-y^2

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simplify log_{e}(-2y*e^{4y-x^2-y^2})

simplify

lo g_{e} (- 2 y \cdot e^{4 y - x^{2} - y^{2}})

ln (2) + ln (y) + 4 y - x^{2} - y^{2}

Solution steps

lo g_{e} (- 2 y e^{4 y - x^{2} - y^{2}})

Apply log rule:

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

lo g_{e} (- 2 y e^{4 y - x^{2} - y^{2}}) = lo g_{e} (2) + lo g_{e} (y) + lo g_{e} (e^{4 y - x^{2} - y^{2}})

= lo g_{e} (2) + lo g_{e} (y) + lo g_{e} (e^{- y^{2} + 4 y - x^{2}})

Apply log rule:

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

lo g_{e} (e^{4 y - x^{2} - y^{2}}) = (4 y - x^{2} - y^{2}) lo g_{e} (e)

= lo g_{e} (2) + lo g_{e} (y) + (4 y - x^{2} - y^{2}) lo g_{e} (e)

lo g_{e} (2) = ln (2)

lo g_{e} (y) = ln (y)

(4 y - x^{2} - y^{2}) lo g_{e} (e) = 4 y - x^{2} - y^{2}

= ln (2) + ln (y) + 4 y - x^{2} - y^{2}

2 lo g_{10} (a) - \frac{1}{2} lo g_{10} (b) + lo g_{10} (c)

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What is log_{e}(-2y*e^{4y-x^2-y^2}) ?
The solution to log_{e}(-2y*e^{4y-x^2-y^2}) is ln(2)+ln(y)+4y-x^2-y^2