What is ln((e^2sqrt(x^2-3))/(x(x+1)^5)) ?

The solution to ln((e^2sqrt(x^2-3))/(x(x+1)^5)) is 2+ln(sqrt(x^2-3))-ln(x)-5ln(x+1)

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simplify ln((e^2sqrt(x^2-3))/(x(x+1)^5))

simplify

ln (\frac{e ^{2} x ^{2} - 3}{x ( x + 1 ) ^{5}})

2 + ln (x^{2} - 3) - ln (x) - 5 ln (x + 1)

Solution steps

ln (\frac{e ^{2} x ^{2} - 3}{x ( x + 1 ) ^{5}})

Apply log rule:

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

ln (\frac{e ^{2} x ^{2} - 3}{x ( x + 1 ) ^{5}}) = ln (e^{2} x^{2} - 3) - ln (x (x + 1)^{5})

= ln (e^{2} x^{2} - 3) - ln (x (x + 1)^{5})

Apply log rule:

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

ln (e^{2} x^{2} - 3) = ln (e^{2}) + ln (x^{2} - 3), ln (x (x + 1)^{5}) = ln (x) + ln ((x + 1)^{5})

= ln (e^{2}) + ln (x^{2} - 3) - (ln (x) + ln ((x + 1)^{5}))

Apply log rule:

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

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

= 2 ln (e) + ln (x^{2} - 3) - (ln (x) + ln ((x + 1)^{5}))

Apply log rule:

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

ln ((x + 1)^{5}) = 5 ln (x + 1)

= 2 ln (e) + ln (x^{2} - 3) - (ln (x) + 5 ln (x + 1))

2 ln (e) = 2

= 2 + ln (x^{2} - 3) - (ln (x) + 5 ln (x + 1))

- (ln (x) + 5 ln (x + 1)) : - ln (x) - 5 ln (x + 1)

= 2 + ln (x^{2} - 3) - ln (x) - 5 ln (x + 1)

\frac{1}{3} ln (x) + \frac{1}{4} ln (y) + c

2 ln (e) + 2 lo g_{x} (+ 3 lo g_{y} (x))

ln (\frac{1}{π})

\frac{1}{2} ln (- 3)

ln (- 23)

What is ln((e^2sqrt(x^2-3))/(x(x+1)^5)) ?
The solution to ln((e^2sqrt(x^2-3))/(x(x+1)^5)) is 2+ln(sqrt(x^2-3))-ln(x)-5ln(x+1)