What is ln((e^{2y}sqrt(xy))/(x^3)) ?

The solution to ln((e^{2y}sqrt(xy))/(x^3)) is 2y+ln(sqrt(yx))-3ln(x)

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simplify ln((e^{2y}sqrt(xy))/(x^3))

simplify

ln (\frac{e ^{2 y} x y}{x ^{3}})

2 y + ln (y x) - 3 ln (x)

Solution steps

ln (\frac{e ^{2 y} x y}{x ^{3}})

Apply log rule:

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

ln (\frac{e ^{2 y} x y}{x ^{3}}) = ln (e^{2 y} x y) - ln (x^{3})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

= 2 y ln (e) + ln (x y) - 3 ln (x)

2 y ln (e) = 2 y

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

\frac{1}{2} ln (\frac{y - 3}{x + 2})

4 ln (2) - 4 ln (2 x)

ln (3) + ln (2) ln (6)

4 ln (x) + 2 ln (y) - 10

20 \cdot lo g_{10} (\frac{10}{2 \cdot 1 0 ^{- 5}})

What is ln((e^{2y}sqrt(xy))/(x^3)) ?
The solution to ln((e^{2y}sqrt(xy))/(x^3)) is 2y+ln(sqrt(yx))-3ln(x)