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

The solution to 1/3 ln((x+2)^3)+1/2 ln(x/((x^2+3x+2)^2)) is ln((sqrt(x))/(x+1))

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

= ln (\frac{x}{x + 1})

- lo g_{10} (1.2 \cdot 1 0^{- 12})

lo g_{4} ((\frac{w ^{4} u ^{4}}{v})^{2})

ln (\frac{395}{373})

\frac{4 ln ( 2 )}{ln ( 10 )} + 2

ln (107109) - ln (109)

What is 1/3 ln((x+2)^3)+1/2 ln(x/((x^2+3x+2)^2)) ?
The solution to 1/3 ln((x+2)^3)+1/2 ln(x/((x^2+3x+2)^2)) is ln((sqrt(x))/(x+1))