What is 1/2 log_{a}(8x^2)-1/6 log_{a}(8x^3) ?

The solution to 1/2 log_{a}(8x^2)-1/6 log_{a}(8x^3) is log_{a}(2sqrt(x))

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simplify 1/2 log_{a}(8x^2)-1/6 log_{a}(8x^3)

simplify

\frac{1}{2} lo g_{a} (8 x^{2}) - \frac{1}{6} lo g_{a} (8 x^{3})

lo g_{a} (2 x)

Solution steps

\frac{1}{2} lo g_{a} (8 x^{2}) - \frac{1}{6} lo g_{a} (8 x^{3})

\frac{1}{2} lo g_{a} (8 x^{2}) = lo g_{a} ((8 x^{2})^{\frac{1}{2}})

- \frac{1}{6} lo g_{a} (8 x^{3}) = - lo g_{a} ((8 x^{3})^{\frac{1}{6}})

= lo g_{a} ((8 x^{2})^{\frac{1}{2}}) - lo g_{a} ((8 x^{3})^{\frac{1}{6}})

Apply log rule:

lo g_{a} (x) - lo g_{a} (y) = lo g_{a} (\frac{x}{y})

lo g_{a} ((8 x^{2})^{\frac{1}{2}}) - lo g_{a} ((8 x^{3})^{\frac{1}{6}}) = lo g_{a} \frac{( 8 x ^{2} ) ^{\frac{1}{2}}}{( 8 x ^{3} ) ^{\frac{1}{6}}}

= lo g_{a} \frac{( 8 x ^{2} ) ^{\frac{1}{2}}}{( 8 x ^{3} ) ^{\frac{1}{6}}}

Simplify

\frac{( 8 x ^{2} ) ^{\frac{1}{2}}}{( 8 x ^{3} ) ^{\frac{1}{6}}} : 2 x^{\frac{1}{2}}

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

Apply exponent rule:

a^{\frac{1}{2}} = a

x^{\frac{1}{2}} = x

= lo g_{a} (2 x)

lo g_{3} (\frac{2 x}{y})

ln (\frac{( 6 x - 4 ) ( 2 y )}{3 x})

lo g_{9} (4) - lo g_{9} (2) + 2 lo g_{9} (5)

lo g_{9} (4) - lo g_{9} (5)

lo g_{10} (\frac{x + 2}{x - 3})

What is 1/2 log_{a}(8x^2)-1/6 log_{a}(8x^3) ?
The solution to 1/2 log_{a}(8x^2)-1/6 log_{a}(8x^3) is log_{a}(2sqrt(x))