What is 1/2*log_{2}(9x^{16})-2/3*log_{2}(8x^3) ?

The solution to 1/2*log_{2}(9x^{16})-2/3*log_{2}(8x^3) is log_{2}((3x^6)/4)

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simplify 1/2log_{2}(9x^{16})-2/3log_{2}(8x^3)

simplify

\frac{1}{2} \cdot lo g_{2} (9 x^{16}) - \frac{2}{3} \cdot lo g_{2} (8 x^{3})

lo g_{2} (\frac{3 x ^{6}}{4})

Solution steps

\frac{1}{2} lo g_{2} (9 x^{16}) - \frac{2}{3} lo g_{2} (8 x^{3})

\frac{1}{2} lo g_{2} (9 x^{16}) = lo g_{2} ((9 x^{16})^{\frac{1}{2}})

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

= lo g_{2} ((9 x^{16})^{\frac{1}{2}}) - lo g_{2} ((8 x^{3})^{\frac{2}{3}})

Apply log rule:

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

lo g_{2} ((9 x^{16})^{\frac{1}{2}}) - lo g_{2} ((8 x^{3})^{\frac{2}{3}}) = lo g_{2} \frac{( 9 x ^{16} ) ^{\frac{1}{2}}}{( 8 x ^{3} ) ^{\frac{2}{3}}}

= lo g_{2} \frac{( 9 x ^{16} ) ^{\frac{1}{2}}}{( 8 x ^{3} ) ^{\frac{2}{3}}}

Simplify

\frac{( 9 x ^{16} ) ^{\frac{1}{2}}}{( 8 x ^{3} ) ^{\frac{2}{3}}} : \frac{3 x ^{6}}{4}

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

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What is 1/2*log_{2}(9x^{16})-2/3*log_{2}(8x^3) ?
The solution to 1/2*log_{2}(9x^{16})-2/3*log_{2}(8x^3) is log_{2}((3x^6)/4)