What is 3/2 ln(4x^8)-2/5 ln(2y^{20}) ?

The solution to 3/2 ln(4x^8)-2/5 ln(2y^{20}) is ln((4*2^{3/5}x^{12})/(y^8))

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simplify 3/2 ln(4x^8)-2/5 ln(2y^{20})

simplify

\frac{3}{2} ln (4 x^{8}) - \frac{2}{5} ln (2 y^{20})

ln (\frac{4 \cdot 2 ^{\frac{3}{5}} x ^{12}}{y ^{8}})

Solution steps

\frac{3}{2} ln (4 x^{8}) - \frac{2}{5} ln (2 y^{20})

Apply log rule:

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

\frac{3}{2} ln (4 x^{8}) = ln ((4 x^{8})^{\frac{3}{2}})

= ln ((4 x^{8})^{\frac{3}{2}}) - \frac{2}{5} ln (2 y^{20})

Apply log rule:

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

\frac{2}{5} ln (2 y^{20}) = ln ((2 y^{20})^{\frac{2}{5}})

= ln ((4 x^{8})^{\frac{3}{2}}) - ln ((2 y^{20})^{\frac{2}{5}})

Apply log rule:

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

ln ((4 x^{8})^{\frac{3}{2}}) - ln ((2 y^{20})^{\frac{2}{5}}) = ln \frac{( 4 x ^{8} ) ^{\frac{3}{2}}}{( 2 y ^{20} ) ^{\frac{2}{5}}}

= ln \frac{( 4 x ^{8} ) ^{\frac{3}{2}}}{( 2 y ^{20} ) ^{\frac{2}{5}}}

Simplify

\frac{( 4 x ^{8} ) ^{\frac{3}{2}}}{( 2 y ^{20} ) ^{\frac{2}{5}}} : \frac{2 ^{2} \cdot 2 ^{\frac{3}{5}} x ^{12}}{y ^{8}}

= ln (\frac{2 ^{2} \cdot 2 ^{\frac{3}{5}} x ^{12}}{y ^{8}})

\frac{2 ^{2} \cdot 2 ^{\frac{3}{5}} x ^{12}}{y ^{8}} = \frac{2 ^{2 + \frac{3}{5}} x ^{12}}{y ^{8}}

= ln (\frac{2 ^{2 + \frac{3}{5}} x ^{12}}{y ^{8}})

\frac{2 ^{2 + \frac{3}{5}} x ^{12}}{y ^{8}} = \frac{4 \cdot 2 ^{\frac{3}{5}} x ^{12}}{y ^{8}}

= ln (\frac{4 \cdot 2 ^{\frac{3}{5}} x ^{12}}{y ^{8}})

2 [ln (x) - ln (x + 5) - ln (x - 5)]

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

ln (2.8 \cdot \frac{( 5 - 1.8 )}{1.8 \cdot ( 5 - 2.8 )})

4 ln ∣ x - 2 ∣ + 10 ln ∣ x + 5 ∣

6 lo g_{3} (u) + 4 lo g_{3} (v)

What is 3/2 ln(4x^8)-2/5 ln(2y^{20}) ?
The solution to 3/2 ln(4x^8)-2/5 ln(2y^{20}) is ln((4*2^{3/5}x^{12})/(y^8))