What is 1/3 ln(x)+10/12 (x+3)-1/6 ln(x-3) ?

The solution to 1/3 ln(x)+10/12 (x+3)-1/6 ln(x-3) is ln((x^{1/3}(x-3)^{5/6}}{x-3})+\frac{5(x+3))/6

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simplify 1/3 ln(x)+10/12 (x+3)-1/6 ln(x-3)

simplify

\frac{1}{3} ln (x) + \frac{10}{12} (x + 3) - \frac{1}{6} ln (x - 3)

ln (\frac{x ^{\frac{1}{3}} ( x - 3 ) ^{\frac{5}{6}}}{x - 3}) + \frac{5 ( x + 3 )}{6}

Solution steps

\frac{1}{3} ln (x) + \frac{10}{12} (x + 3) - \frac{1}{6} ln (x - 3)

Apply log rule:

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

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

= ln (x^{\frac{1}{3}}) + \frac{10}{12} (x + 3) - \frac{1}{6} ln (x - 3)

Apply log rule:

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

\frac{1}{6} ln (x - 3) = ln ((x - 3)^{\frac{1}{6}})

= ln (x^{\frac{1}{3}}) + \frac{10}{12} (x + 3) - ln ((x - 3)^{\frac{1}{6}})

Apply log rule:

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

ln (x^{\frac{1}{3}}) - ln ((x - 3)^{\frac{1}{6}}) = ln (\frac{x ^{\frac{1}{3}}}{( x - 3 ) ^{\frac{1}{6}}})

= ln (\frac{x ^{\frac{1}{3}}}{( x - 3 ) ^{\frac{1}{6}}}) + \frac{10}{12} (x + 3)

Simplify

\frac{x ^{\frac{1}{3}}}{( x - 3 ) ^{\frac{1}{6}}} : \frac{x ^{\frac{1}{3}} ( x - 3 ) ^{\frac{5}{6}}}{x - 3}

= ln (\frac{x ^{\frac{1}{3}} ( x - 3 ) ^{\frac{5}{6}}}{x - 3}) + \frac{10}{12} (x + 3)

\frac{10}{12} (x + 3) = \frac{5 ( x + 3 )}{6}

= ln (\frac{x ^{\frac{1}{3}} ( x - 3 ) ^{\frac{5}{6}}}{x - 3}) + \frac{5 ( x + 3 )}{6}

\frac{1}{2} ln (5) - ln (2)

4.74 + lo g_{10} (\frac{0.81245}{0.24745})

ln (1300) - ln (100 \cdot e^{0.09})

4 lo g_{6} (x) + 5 lo g_{6} (y)

What is 1/3 ln(x)+10/12 (x+3)-1/6 ln(x-3) ?
The solution to 1/3 ln(x)+10/12 (x+3)-1/6 ln(x-3) is ln((x^{1/3}(x-3)^{5/6}}{x-3})+\frac{5(x+3))/6