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

The solution to 2ln(2)-1/2 ln(x^3+3x) is ln((4sqrt(x^3+3x))/(x^3+3x))

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

4 ln (2 cos (x))

10 - 2 lo g_{4} (x) - 4 lo g_{4} (y)

lo g_{10} (\frac{x ^{2}}{3})

\frac{1}{2} lo g_{b} (\frac{3 + 2 2}{3 - 2 2})

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

What is 2ln(2)-1/2 ln(x^3+3x) ?
The solution to 2ln(2)-1/2 ln(x^3+3x) is ln((4sqrt(x^3+3x))/(x^3+3x))