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

The solution to 6ln|x+1|+3ln|x-3| is ln(|x-3|^3(x+1)^6)

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

simplify

6 ln ∣ x + 1 ∣ + 3 ln ∣ x - 3 ∣

ln (∣ x - 3 ∣^{3} (x + 1)^{6})

Solution steps

6 ln ∣ x + 1 ∣ + 3 ln ∣ x - 3 ∣

Apply log rule:

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

6 ln ∣ x + 1 ∣ = ln (∣ x + 1 ∣^{6})

= ln (∣ x + 1 ∣^{6}) + 3 ln ∣ x - 3 ∣

Apply log rule:

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

3 ln ∣ x - 3 ∣ = ln (∣ x - 3 ∣^{3})

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

Apply log rule:

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

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

= ln (∣ x + 1 ∣^{6} ∣ x - 3 ∣^{3})

∣ x + 1 ∣^{6} = (x + 1)^{6}

= ln (∣ x - 3 ∣^{3} (x + 1)^{6})

ln (6 \cdot 1 0^{6})

ln (a) - 5 ln (b) + 3 ln (c)

ln (\frac{2 3}{3})

6 ln (x - 5) - ln (x^{2} - 8 x + 15) + 2 ln (x - 3)

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

What is 6ln|x+1|+3ln|x-3| ?
The solution to 6ln|x+1|+3ln|x-3| is ln(|x-3|^3(x+1)^6)