What is ln(x)-2ln(x^2+1)+1/2 ln(3-x^4) ?

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

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simplify ln(x)-2ln(x^2+1)+1/2 ln(3-x^4)

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

lo g_{b} (\frac{p ^{4} q ^{7}}{m ^{5} b ^{7}})

lo g_{10} (y z^{6})

lo g_{10} (3 x^{2} y)

ln (\frac{23}{0.22})

4 lo g_{10} (x) + lo g_{10} (x^{2}) - lo g_{10} (x^{4})

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