What is 1/6 ln|x^2+9|+1/6 ln|(x^2)/9+1| ?

The solution to 1/6 ln|x^2+9|+1/6 ln|(x^2)/9+1| is ln(|x^2+9|^{1/6}|(x^2)/9+1|^{1/6})

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simplify 1/6 ln|x^2+9|+1/6 ln|(x^2)/9+1|

simplify

\frac{1}{6} ln x^{2} + 9 + \frac{1}{6} ln \frac{x ^{2}}{9} + 1

ln (x^{2} + 9^{\frac{1}{6}} \frac{x ^{2}}{9} + 1^{\frac{1}{6}})

Solution steps

\frac{1}{6} ln x^{2} + 9 + \frac{1}{6} ln \frac{x ^{2}}{9} + 1

Apply log rule:

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

\frac{1}{6} ln x^{2} + 9 = ln (x^{2} + 9^{\frac{1}{6}})

= ln (x^{2} + 9^{\frac{1}{6}}) + \frac{1}{6} ln \frac{x ^{2}}{9} + 1

Apply log rule:

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

\frac{1}{6} ln \frac{x ^{2}}{9} + 1 = ln (\frac{x ^{2}}{9} + 1^{\frac{1}{6}})

= ln (x^{2} + 9^{\frac{1}{6}}) + ln (\frac{x ^{2}}{9} + 1^{\frac{1}{6}})

Apply log rule:

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

ln (x^{2} + 9^{\frac{1}{6}}) + ln (\frac{x ^{2}}{9} + 1^{\frac{1}{6}}) = ln (x^{2} + 9^{\frac{1}{6}} \frac{x ^{2}}{9} + 1^{\frac{1}{6}})

= ln (x^{2} + 9^{\frac{1}{6}} \frac{x ^{2}}{9} + 1^{\frac{1}{6}})

ln (\frac{x ^{4}}{y ^{5}})

lo g_{3} (\frac{a ^{5} b}{c ^{2}})

2 ln (5) - 2 ln (3)

lo g_{2} (x - 4) + 5 lo g_{2} (x + 1) - 3 lo g_{2} (x - 1)

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What is 1/6 ln|x^2+9|+1/6 ln|(x^2)/9+1| ?
The solution to 1/6 ln|x^2+9|+1/6 ln|(x^2)/9+1| is ln(|x^2+9|^{1/6}|(x^2)/9+1|^{1/6})