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

The solution to 1/3 ln(x)+10/12 ln(x+3)-1/6 ln(x+3) is ln(x^{1/3}(x+3)^{2/3})

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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

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

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

Simplify

\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}}}{( x + 3 ) ^{\frac{1}{6}}} : x^{\frac{1}{3}} (x + 3)^{\frac{2}{3}}

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

ln (\frac{350}{400 - x})

- ln (2) + ln (11) - ln (10) - ln (55)

ln (\frac{3.64}{23}) + \frac{3}{23} ln (23)

\frac{1}{3} ln (\frac{17}{3} + 35)

\frac{1}{3} (ln (x) - ln (y))

What is 1/3 ln(x)+10/12 ln(x+3)-1/6 ln(x+3) ?
The solution to 1/3 ln(x)+10/12 ln(x+3)-1/6 ln(x+3) is ln(x^{1/3}(x+3)^{2/3})