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

The solution to 2ln(3)-1/2 ln(x^2+1) is ln((9sqrt(x^2+1))/(x^2+1))

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

simplify

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

ln (\frac{9 x ^{2} + 1}{x ^{2} + 1})

Solution steps

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

Apply log rule:

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

2 ln (3) = ln (3^{2})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

\frac{3 ^{2}}{( x ^{2} + 1 ) ^{\frac{1}{2}}} : \frac{9 x ^{2} + 1}{x ^{2} + 1}

= ln (\frac{9 x ^{2} + 1}{x ^{2} + 1})

lo g_{3} (9 x^{2})

\frac{1}{3} lo g_{b} (x) + 5 lo g_{b} (y) - 3 lo g_{b} (x)

lo g_{5} (\frac{12 x}{3 y})

lo g_{10} (16) + 2 lo g_{10} (3)

ln (\frac{10}{5})

What is 2ln(3)-1/2 ln(x^2+1) ?
The solution to 2ln(3)-1/2 ln(x^2+1) is ln((9sqrt(x^2+1))/(x^2+1))