What is 2(-ln|x+3|+2ln|x+6|)+C ?

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

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simplify 2(-ln|x+3|+2ln|x+6|)+C

simplify

2 (- ln ∣ x + 3 ∣ + 2 ln ∣ x + 6 ∣) + C

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

Solution steps

2 (- ln ∣ x + 3 ∣ + 2 ln ∣ x + 6 ∣) + C

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

2 ln ∣ x + 6 ∣ = ln (∣ x + 6 ∣^{2})

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

Apply log rule:

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

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

= 2 (ln (\frac{∣ x + 6 ∣ ^{2}}{∣ x + 3 ∣})) + C

∣ x + 6 ∣^{2} = (x + 6)^{2}

= 2 (ln (\frac{( x + 6 ) ^{2}}{∣ x + 3 ∣})) + C

Remove parentheses:

(a) = a

= 2 ln (\frac{( x + 6 ) ^{2}}{∣ x + 3 ∣}) + C

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