What is ln((sqrt(x))/(e^x(x+3)^2)) ?

The solution to ln((sqrt(x))/(e^x(x+3)^2)) is ln(sqrt(x))-x-2ln(x+3)

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simplify ln((sqrt(x))/(e^x(x+3)^2))

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

ln ((x + 3)^{2}) = 2 ln (x + 3)

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

x ln (e) = x

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

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

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

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

ln (\frac{2}{e ^{(- 1)} + e})

\frac{1}{2} ln (\frac{π}{4}) + c

ln (\frac{2 ^{- 1}}{3 ^{2} \cdot 5 ^{1}})

ln (9) + 2 ln (4) - ln (2)

What is ln((sqrt(x))/(e^x(x+3)^2)) ?
The solution to ln((sqrt(x))/(e^x(x+3)^2)) is ln(sqrt(x))-x-2ln(x+3)