What is ln(((x+5)(x+6)^6)/((x+2)^2(x+3)^3)) ?

The solution to ln(((x+5)(x+6)^6)/((x+2)^2(x+3)^3)) is ln(x+5)+6ln(x+6)-2ln(x+2)-3ln(x+3)

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simplify ln(((x+5)(x+6)^6)/((x+2)^2(x+3)^3))

simplify

ln (\frac{( x + 5 ) ( x + 6 ) ^{6}}{( x + 2 ) ^{2} ( x + 3 ) ^{3}})

ln (x + 5) + 6 ln (x + 6) - 2 ln (x + 2) - 3 ln (x + 3)

Solution steps

ln (\frac{( x + 5 ) ( x + 6 ) ^{6}}{( x + 2 ) ^{2} ( x + 3 ) ^{3}})

Apply log rule:

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

ln (\frac{( x + 5 ) ( x + 6 ) ^{6}}{( x + 2 ) ^{2} ( x + 3 ) ^{3}}) = ln ((x + 5) (x + 6)^{6}) - ln ((x + 2)^{2} (x + 3)^{3})

= ln ((x + 5) (x + 6)^{6}) - ln ((x + 2)^{2} (x + 3)^{3})

Apply log rule:

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

ln ((x + 5) (x + 6)^{6}) = ln (x + 5) + ln ((x + 6)^{6}), ln ((x + 2)^{2} (x + 3)^{3}) = ln ((x + 2)^{2}) + ln ((x + 3)^{3})

= ln (x + 5) + ln ((x + 6)^{6}) - (ln ((x + 2)^{2}) + ln ((x + 3)^{3}))

Apply log rule:

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

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

= ln (x + 5) + 6 ln (x + 6) - (ln ((x + 2)^{2}) + ln ((x + 3)^{3}))

Apply log rule:

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

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

= ln (x + 5) + 6 ln (x + 6) - (2 ln (x + 2) + ln ((x + 3)^{3}))

Apply log rule:

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

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

= ln (x + 5) + 6 ln (x + 6) - (2 ln (x + 2) + 3 ln (x + 3))

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

= ln (x + 5) + 6 ln (x + 6) - 2 ln (x + 2) - 3 ln (x + 3)

ln (\frac{1}{1 0 ^{- 5}})

- lo g_{10} (5.41295 \cdot 1 0^{- 8})

ln (3 x) + ln (4 x)

- lo g_{10} (3.3 \cdot 1 0^{- 1})

\frac{\partial}{\partial x} (x y) + 2 x - ln (x^{2} y)

What is ln(((x+5)(x+6)^6)/((x+2)^2(x+3)^3)) ?
The solution to ln(((x+5)(x+6)^6)/((x+2)^2(x+3)^3)) is ln(x+5)+6ln(x+6)-2ln(x+2)-3ln(x+3)