What is 3ln|x-1|-1/3 ln|3x-6|+c ?

The solution to 3ln|x-1|-1/3 ln|3x-6|+c is ln((|x-1|^3|3x-6|^{2/3})/(|3x-6|))+c

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simplify 3ln|x-1|-1/3 ln|3x-6|+c

simplify

3 ln ∣ x - 1 ∣ - \frac{1}{3} ln ∣ 3 x - 6 ∣ + c

ln (\frac{∣ x - 1 ∣ ^{3} ∣ 3 x - 6 ∣ ^{\frac{2}{3}}}{∣ 3 x - 6 ∣}) + c

Solution steps

3 ln ∣ x - 1 ∣ - \frac{1}{3} ln ∣ 3 x - 6 ∣ + c

Apply log rule:

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

3 ln ∣ x - 1 ∣ = ln (∣ x - 1 ∣^{3})

= ln (∣ x - 1 ∣^{3}) - \frac{1}{3} ln ∣ 3 x - 6 ∣ + c

Apply log rule:

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

\frac{1}{3} ln ∣ 3 x - 6 ∣ = ln (∣ 3 x - 6 ∣^{\frac{1}{3}})

= ln (∣ x - 1 ∣^{3}) - ln (∣ 3 x - 6 ∣^{\frac{1}{3}}) + c

Apply log rule:

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

ln (∣ x - 1 ∣^{3}) - ln (∣ 3 x - 6 ∣^{\frac{1}{3}}) = ln (\frac{∣ x - 1 ∣ ^{3}}{∣ 3 x - 6 ∣ ^{\frac{1}{3}}})

= ln (\frac{∣ x - 1 ∣ ^{3}}{∣ 3 x - 6 ∣ ^{\frac{1}{3}}}) + c

Simplify

\frac{∣ x - 1 ∣ ^{3}}{∣ 3 x - 6 ∣ ^{\frac{1}{3}}} : \frac{∣ x - 1 ∣ ^{3} ∣ 3 x - 6 ∣ ^{\frac{2}{3}}}{∣ 3 x - 6 ∣}

= ln (\frac{∣ x - 1 ∣ ^{3} ∣ 3 x - 6 ∣ ^{\frac{2}{3}}}{∣ 3 x - 6 ∣}) + c

ln (3) (- 1 - \frac{1}{2} - \frac{1}{8} + \frac{3}{4}) + ln (2) (- 1 - \frac{1}{4} - \frac{3}{4})

ln (x) - ln (3 x)

ln (\frac{1}{1 - 0.97})

ln (\frac{e ^{8} - 1}{e ^{2} - 1}) - 3

4 ln (2 x)

What is 3ln|x-1|-1/3 ln|3x-6|+c ?
The solution to 3ln|x-1|-1/3 ln|3x-6|+c is ln((|x-1|^3|3x-6|^{2/3})/(|3x-6|))+c