What is ln|x|+3/2 ln|x^2+6| ?

The solution to ln|x|+3/2 ln|x^2+6| is ln(|x||x^2+6|^{3/2})

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simplify ln|x|+3/2 ln|x^2+6|

simplify

ln ∣ x ∣ + \frac{3}{2} ln x^{2} + 6

ln (∣ x ∣ x^{2} + 6^{\frac{3}{2}})

Solution steps

ln ∣ x ∣ + \frac{3}{2} ln x^{2} + 6

Apply log rule:

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

\frac{3}{2} ln x^{2} + 6 = ln (x^{2} + 6^{\frac{3}{2}})

= ln ∣ x ∣ + ln (x^{2} + 6^{\frac{3}{2}})

Apply log rule:

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

ln ∣ x ∣ + ln (x^{2} + 6^{\frac{3}{2}}) = ln (∣ x ∣ x^{2} + 6^{\frac{3}{2}})

= ln (∣ x ∣ x^{2} + 6^{\frac{3}{2}})

\frac{57}{40} + \frac{5}{4} ln (\frac{4}{5}) + \frac{3}{8} ln (2)

lo g_{7} (\frac{1 + 2 e ^{- x}}{e ^{2 x}})

ln ((24 e^{- x - y})^{5}) + ln ((15 e^{3 x - y})^{3})

ln (\frac{0.85}{0.6108})

lo g_{3} ((6 x)^{2})

What is ln|x|+3/2 ln|x^2+6| ?
The solution to ln|x|+3/2 ln|x^2+6| is ln(|x||x^2+6|^{3/2})