What is 1/3 ln(x)+4ln(((x-3)(x+3)^{5/8})/(x+3)) ?

The solution to 1/3 ln(x)+4ln(((x-3)(x+3)^{5/8})/(x+3)) is ln((x^{1/3}(x-3)^4)/((x+3)^{3/2)})

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simplify 1/3 ln(x)+4ln(((x-3)(x+3)^{5/8})/(x+3))

simplify

\frac{1}{3} ln (x) + 4 ln (\frac{( x - 3 ) ( x + 3 ) ^{\frac{5}{8}}}{x + 3})

ln (\frac{x ^{\frac{1}{3}} ( x - 3 ) ^{4}}{( x + 3 ) ^{\frac{3}{2}}})

Solution steps

\frac{1}{3} ln (x) + 4 ln (\frac{( x - 3 ) ( x + 3 ) ^{\frac{5}{8}}}{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}}) + 4 ln (\frac{( x - 3 ) ( x + 3 ) ^{\frac{5}{8}}}{x + 3})

Apply log rule:

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

4 ln (\frac{( x - 3 ) ( x + 3 ) ^{\frac{5}{8}}}{x + 3}) = ln (\frac{( x - 3 ) ( x + 3 ) ^{\frac{5}{8}}}{x + 3})^{4}

= ln (x^{\frac{1}{3}}) + ln (\frac{( x - 3 ) ( x + 3 ) ^{\frac{5}{8}}}{x + 3})^{4}

Apply log rule:

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

ln (x^{\frac{1}{3}}) + ln (\frac{( x - 3 ) ( x + 3 ) ^{\frac{5}{8}}}{x + 3})^{4} = ln (\frac{( x - 3 ) ( x + 3 ) ^{\frac{5}{8}}}{x + 3})^{4} x^{\frac{1}{3}}

= ln (\frac{( x - 3 ) ( x + 3 ) ^{\frac{5}{8}}}{x + 3})^{4} x^{\frac{1}{3}}

Simplify

x^{\frac{1}{3}} (\frac{( x - 3 ) ( x + 3 ) ^{\frac{5}{8}}}{x + 3})^{4} : \frac{x ^{\frac{1}{3}} ( x - 3 ) ^{4}}{( x + 3 ) ^{\frac{3}{2}}}

= ln (\frac{x ^{\frac{1}{3}} ( x - 3 ) ^{4}}{( x + 3 ) ^{\frac{3}{2}}})

4 lo g_{4} (2) + 2 lo g_{4} (7)

lo g_{10} (\frac{s}{( s - 1 )})

lo g_{10} (2 a b)

lo g_{10} ((x - y)^{2}) - lo g_{10} (x - y)

ln (- 6 (\frac{e ^{1}}{3})^{\frac{1}{2}} \cdot 1)

What is 1/3 ln(x)+4ln(((x-3)(x+3)^{5/8})/(x+3)) ?
The solution to 1/3 ln(x)+4ln(((x-3)(x+3)^{5/8})/(x+3)) is ln((x^{1/3}(x-3)^4)/((x+3)^{3/2)})