What is ln(((x^5sqrt(x^2+1)))/((x+1)^6)) ?

The solution to ln(((x^5sqrt(x^2+1)))/((x+1)^6)) is 5ln(x)+ln(sqrt(x^2+1))-6ln(x+1)

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simplify ln(((x^5sqrt(x^2+1)))/((x+1)^6))

simplify

ln (\frac{( x ^{5} x ^{2} + 1 )}{( x + 1 ) ^{6}})

5 ln (x) + ln (x^{2} + 1) - 6 ln (x + 1)

Solution steps

ln (\frac{( x ^{5} x ^{2} + 1 )}{( x + 1 ) ^{6}})

Apply log rule:

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

ln (\frac{( x ^{5} x ^{2} + 1 )}{( x + 1 ) ^{6}}) = ln ((x^{5} x^{2} + 1)) - ln ((x + 1)^{6})

= ln (x^{5} x^{2} + 1) - ln ((x + 1)^{6})

Apply log rule:

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

ln ((x + 1)^{6}) = 6 ln (x + 1)

= ln (x^{5} x^{2} + 1) - 6 ln (x + 1)

Apply log rule:

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

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

= ln (x^{5}) + ln (x^{2} + 1) - 6 ln (x + 1)

Apply log rule:

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

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

= 5 ln (x) + ln (x^{2} + 1) - 6 ln (x + 1)

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What is ln(((x^5sqrt(x^2+1)))/((x+1)^6)) ?
The solution to ln(((x^5sqrt(x^2+1)))/((x+1)^6)) is 5ln(x)+ln(sqrt(x^2+1))-6ln(x+1)