What is 1/3 ln|x-3|+1/3 ln|x^2+9| ?

The solution to 1/3 ln|x-3|+1/3 ln|x^2+9| is ln(|x-3|^{1/3}|x^2+9|^{1/3})

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simplify 1/3 ln|x-3|+1/3 ln|x^2+9|

simplify

\frac{1}{3} ln ∣ x - 3 ∣ + \frac{1}{3} ln x^{2} + 9

ln (∣ x - 3 ∣^{\frac{1}{3}} x^{2} + 9^{\frac{1}{3}})

Solution steps

\frac{1}{3} ln ∣ x - 3 ∣ + \frac{1}{3} ln x^{2} + 9

Apply log rule:

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

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

= ln (∣ x - 3 ∣^{\frac{1}{3}}) + \frac{1}{3} ln x^{2} + 9

Apply log rule:

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

\frac{1}{3} ln x^{2} + 9 = ln (x^{2} + 9^{\frac{1}{3}})

= ln (∣ x - 3 ∣^{\frac{1}{3}}) + ln (x^{2} + 9^{\frac{1}{3}})

Apply log rule:

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

ln (∣ x - 3 ∣^{\frac{1}{3}}) + ln (x^{2} + 9^{\frac{1}{3}}) = ln (∣ x - 3 ∣^{\frac{1}{3}} x^{2} + 9^{\frac{1}{3}})

= ln (∣ x - 3 ∣^{\frac{1}{3}} x^{2} + 9^{\frac{1}{3}})

ln (2) ln (\frac{2 y}{0.5})

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What is 1/3 ln|x-3|+1/3 ln|x^2+9| ?
The solution to 1/3 ln|x-3|+1/3 ln|x^2+9| is ln(|x-3|^{1/3}|x^2+9|^{1/3})