What is 1/3 ln(x)+10/12 ln(x+3)-1/16 ln(x-3) ?

The solution to 1/3 ln(x)+10/12 ln(x+3)-1/16 ln(x-3) is ln((x^{1/3}(x+3)^{5/6}(x-3)^{15/16})/(x-3))

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simplify 1/3 ln(x)+10/12 ln(x+3)-1/16 ln(x-3)

simplify

\frac{1}{3} ln (x) + \frac{10}{12} ln (x + 3) - \frac{1}{16} ln (x - 3)

ln (\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}} ( x - 3 ) ^{\frac{15}{16}}}{x - 3})

Solution steps

\frac{1}{3} ln (x) + \frac{10}{12} ln (x + 3) - \frac{1}{16} 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}{16} 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}{16} 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}{16} 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}{16} ln (x - 3)

Apply log rule:

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

\frac{1}{16} ln (x - 3) = ln ((x - 3)^{\frac{1}{16}})

= ln (x^{\frac{1}{3}} (x + 3)^{\frac{5}{6}}) - ln ((x - 3)^{\frac{1}{16}})

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}{16}}) = ln (\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}}}{( x - 3 ) ^{\frac{1}{16}}})

= ln (\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}}}{( x - 3 ) ^{\frac{1}{16}}})

Simplify

\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}}}{( x - 3 ) ^{\frac{1}{16}}} : \frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}} ( x - 3 ) ^{\frac{15}{16}}}{x - 3}

= ln (\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}} ( x - 3 ) ^{\frac{15}{16}}}{x - 3})

lo g_{5} (\frac{c}{125})

2000 (ln (140000) - ln (122412.5)) - 82.075

ln (2) - ln (x + 2)

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

- lo g_{10} (6.667 \cdot 1 0^{- 5})

What is 1/3 ln(x)+10/12 ln(x+3)-1/16 ln(x-3) ?
The solution to 1/3 ln(x)+10/12 ln(x+3)-1/16 ln(x-3) is ln((x^{1/3}(x+3)^{5/6}(x-3)^{15/16})/(x-3))