What is 2log_{a}(4x^2)-1/6 log_{a}(2x+3) ?

The solution to 2log_{a}(4x^2)-1/6 log_{a}(2x+3) is log_{a}((16x^4(2x+3)^{5/6})/(2x+3))

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simplify 2log_{a}(4x^2)-1/6 log_{a}(2x+3)

simplify

2 lo g_{a} (4 x^{2}) - \frac{1}{6} lo g_{a} (2 x + 3)

lo g_{a} (\frac{16 x ^{4} ( 2 x + 3 ) ^{\frac{5}{6}}}{2 x + 3})

Solution steps

2 lo g_{a} (4 x^{2}) - \frac{1}{6} lo g_{a} (2 x + 3)

Apply log rule:

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

2 lo g_{a} (4 x^{2}) = lo g_{a} ((4 x^{2})^{2})

= lo g_{a} ((4 x^{2})^{2}) - \frac{1}{6} lo g_{a} (2 x + 3)

Apply log rule:

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

\frac{1}{6} lo g_{a} (2 x + 3) = lo g_{a} ((2 x + 3)^{\frac{1}{6}})

= lo g_{a} ((4 x^{2})^{2}) - lo g_{a} ((2 x + 3)^{\frac{1}{6}})

Apply log rule:

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

lo g_{a} ((4 x^{2})^{2}) - lo g_{a} ((2 x + 3)^{\frac{1}{6}}) = lo g_{a} (\frac{( 4 x ^{2} ) ^{2}}{( 2 x + 3 ) ^{\frac{1}{6}}})

= lo g_{a} (\frac{( 4 x ^{2} ) ^{2}}{( 2 x + 3 ) ^{\frac{1}{6}}})

Simplify

\frac{( 4 x ^{2} ) ^{2}}{( 2 x + 3 ) ^{\frac{1}{6}}} : \frac{16 x ^{4} ( 2 x + 3 ) ^{\frac{5}{6}}}{2 x + 3}

= lo g_{a} (\frac{16 x ^{4} ( 2 x + 3 ) ^{\frac{5}{6}}}{2 x + 3})

- lo g_{10} (1.66 \cdot 1 0^{(- 5)})

\frac{1}{2} (ln (8) - ln (3))

lo g_{π} (\frac{22}{7})

ln (\frac{2000}{0.0012 \cdot ( - 5 \cdot 1 0 ^{16} )})

\frac{1}{10} ln (\frac{70}{80})

What is 2log_{a}(4x^2)-1/6 log_{a}(2x+3) ?
The solution to 2log_{a}(4x^2)-1/6 log_{a}(2x+3) is log_{a}((16x^4(2x+3)^{5/6})/(2x+3))