What is ln((x^8sqrt(x^2+2))/((x+2)^9)) ?

The solution to ln((x^8sqrt(x^2+2))/((x+2)^9)) is 8ln(x)+ln(sqrt(x^2+2))-9ln(x+2)

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simplify ln((x^8sqrt(x^2+2))/((x+2)^9))

simplify

ln (\frac{x ^{8} x ^{2} + 2}{( x + 2 ) ^{9}})

8 ln (x) + ln (x^{2} + 2) - 9 ln (x + 2)

Solution steps

ln (\frac{x ^{8} x ^{2} + 2}{( x + 2 ) ^{9}})

Apply log rule:

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

ln (\frac{x ^{8} x ^{2} + 2}{( x + 2 ) ^{9}}) = ln (x^{8} x^{2} + 2) - ln ((x + 2)^{9})

= ln (x^{8} x^{2} + 2) - ln ((x + 2)^{9})

Apply log rule:

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

ln ((x + 2)^{9}) = 9 ln (x + 2)

= ln (x^{8} x^{2} + 2) - 9 ln (x + 2)

Apply log rule:

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

ln (x^{8} x^{2} + 2) = ln (x^{8}) + ln (x^{2} + 2)

= ln (x^{8}) + ln (x^{2} + 2) - 9 ln (x + 2)

Apply log rule:

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

ln (x^{8}) = 8 ln (x)

= 8 ln (x) + ln (x^{2} + 2) - 9 ln (x + 2)

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What is ln((x^8sqrt(x^2+2))/((x+2)^9)) ?
The solution to ln((x^8sqrt(x^2+2))/((x+2)^9)) is 8ln(x)+ln(sqrt(x^2+2))-9ln(x+2)