What is 2/3 ln(3-x)-1/3 ln(3+x) ?

The solution to 2/3 ln(3-x)-1/3 ln(3+x) is ln(\sqrt[3]{((3-x)^2)/(3+x)})

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simplify 2/3 ln(3-x)-1/3 ln(3+x)

simplify

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

Solution steps

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

Apply log rule:

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

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

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

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

Apply log rule:

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

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

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

Combine same powers :

- lo g_{10} (6 \cdot 1 0^{- 7})

ln (\frac{6.5}{2.5})

ln (\frac{π}{4}) \cdot \frac{π}{4}

3 ln (17 x^{2})

ln (\frac{3.64}{23}) + \frac{2}{23} ln (23)

What is 2/3 ln(3-x)-1/3 ln(3+x) ?
The solution to 2/3 ln(3-x)-1/3 ln(3+x) is ln(\sqrt[3]{((3-x)^2)/(3+x)})