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}})

\frac{1}{5} ln (5 ∣ ln (x) ∣) + C

2 ln (3) - ln (5) - \frac{16}{45}

\frac{1}{6} \cdot lo g_{2} (\frac{1}{6}) + \frac{5}{6} \cdot lo g_{2} (\frac{5}{6})

ln \frac{( \frac{7 + 3 5}{2} )}{6}

ln (\frac{1}{96})

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})