What is ln((x^3sqrt(x^2+1))/((x+1)^4)) ?

The solution to ln((x^3sqrt(x^2+1))/((x+1)^4)) is 3ln(x)+ln(sqrt(x^2+1))-4ln(x+1)

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simplify ln((x^3sqrt(x^2+1))/((x+1)^4))

simplify

ln (\frac{x ^{3} x ^{2} + 1}{( x + 1 ) ^{4}})

3 ln (x) + ln (x^{2} + 1) - 4 ln (x + 1)

Solution steps

ln (\frac{x ^{3} x ^{2} + 1}{( x + 1 ) ^{4}})

Apply log rule:

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

ln (\frac{x ^{3} x ^{2} + 1}{( x + 1 ) ^{4}}) = ln (x^{3} x^{2} + 1) - ln ((x + 1)^{4})

= ln (x^{3} x^{2} + 1) - ln ((x + 1)^{4})

Apply log rule:

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

ln ((x + 1)^{4}) = 4 ln (x + 1)

= ln (x^{3} x^{2} + 1) - 4 ln (x + 1)

Apply log rule:

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

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

= ln (x^{3}) + ln (x^{2} + 1) - 4 ln (x + 1)

Apply log rule:

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

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

= 3 ln (x) + ln (x^{2} + 1) - 4 ln (x + 1)

- ln (5) + 4 ln (2) - ln (3)

lo g_{7} (4 x)

\frac{1}{3} lo g_{c} (x) + 5 lo g_{c} (y) - 2 lo g_{c} (x)

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

What is ln((x^3sqrt(x^2+1))/((x+1)^4)) ?
The solution to ln((x^3sqrt(x^2+1))/((x+1)^4)) is 3ln(x)+ln(sqrt(x^2+1))-4ln(x+1)