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

The solution to 3ln(4)-1/2 ln(x^2+3) is ln((64sqrt(x^2+3))/(x^2+3))

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

simplify

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

ln (\frac{64 x ^{2} + 3}{x ^{2} + 3})

Solution steps

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

Apply log rule:

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

3 ln (4) = ln (4^{3})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

\frac{4 ^{3}}{( x ^{2} + 3 ) ^{\frac{1}{2}}} : \frac{64 x ^{2} + 3}{x ^{2} + 3}

= ln (\frac{64 x ^{2} + 3}{x ^{2} + 3})

\frac{3}{5} lo g_{\frac{4}{25}} (5) - \frac{3}{5} lo g_{\frac{4}{25}} (2)

\frac{3}{4} \cdot lo g_{2} (\frac{( \frac{3}{4} )}{( \frac{1}{20} )})

\frac{8}{π} \cdot ln (6) + \frac{2}{π} \cdot ln (\frac{7}{6})

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

0.0004 ln ∣ 2000 ∣ - 0.004 ln ∣ 2000 - 25000 ∣

What is 3ln(4)-1/2 ln(x^2+3) ?
The solution to 3ln(4)-1/2 ln(x^2+3) is ln((64sqrt(x^2+3))/(x^2+3))