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

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

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simplify 2ln(x)+xln(3)-1/2 ln(x+1)

simplify

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

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

Solution steps

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

Apply log rule:

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

2 ln (x) = ln (x^{2})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

ln (\frac{4}{2})

lo g_{10} (\frac{x y}{z})

lo g_{2} (\frac{8}{x + 8})

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