What is x^2-1/2 ln|x+2|+3/2 ln|x-4|+C ?

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

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simplify x^2-1/2 ln|x+2|+3/2 ln|x-4|+C

simplify

x^{2} - \frac{1}{2} ln ∣ x + 2 ∣ + \frac{3}{2} ln ∣ x - 4 ∣ + C

x^{2} + ln \frac{∣ x - 4 ∣ ^{3}}{∣ x + 2 ∣} + C

Solution steps

x^{2} - \frac{1}{2} ln ∣ x + 2 ∣ + \frac{3}{2} ln ∣ x - 4 ∣ + C

Apply log rule:

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

\frac{1}{2} ln ∣ x + 2 ∣ = ln (∣ x + 2 ∣^{\frac{1}{2}})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Combine same powers :

= x^{2} + ln \frac{∣ x - 4 ∣ ^{3}}{∣ x + 2 ∣} + C

\frac{1}{12} (lo g_{10} (\frac{3}{4}) + lo g_{10} (\frac{5}{4}))

lo g_{6} (18) + lo g_{6} (3)

lo g_{10} (1 0^{6} x^{2})

ln (a) + ln (b) - (ln (a) + ln (c))

2 ln (5) - ln (2)

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