sendo log de a na base 3= r, log de b na base 3 = s, e log de c na base 3 = t, obtenha log3 a2 .raiz de b/raiz cubica de c, em funçao de r,s,t.
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![\boxed{\boxed{\sqrt[n]{a^{m}}=a^{m/n}}}\\\\\\\boxed{\boxed{log_{b}(x\cdot y)=log_{b}(x)+log_{b}(y)}}\\\\\\\boxed{\boxed{log_{b}\left(\dfrac{x}{y}\right)=log_{b}(x)-log_{b}(y)}}\\\\\\\boxed{\boxed{log_{b}(a^{n})=n\cdot log_{b}(a)}}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cboxed%7B%5Csqrt%5Bn%5D%7Ba%5E%7Bm%7D%7D%3Da%5E%7Bm%2Fn%7D%7D%7D%5C%5C%5C%5C%5C%5C%5Cboxed%7B%5Cboxed%7Blog_%7Bb%7D%28x%5Ccdot+y%29%3Dlog_%7Bb%7D%28x%29%2Blog_%7Bb%7D%28y%29%7D%7D%5C%5C%5C%5C%5C%5C%5Cboxed%7B%5Cboxed%7Blog_%7Bb%7D%5Cleft%28%5Cdfrac%7Bx%7D%7By%7D%5Cright%29%3Dlog_%7Bb%7D%28x%29-log_%7Bb%7D%28y%29%7D%7D%5C%5C%5C%5C%5C%5C%5Cboxed%7B%5Cboxed%7Blog_%7Bb%7D%28a%5E%7Bn%7D%29%3Dn%5Ccdot+log_%7Bb%7D%28a%29%7D%7D "\boxed{\boxed{\sqrt[n]{a^{m}}=a^{m/n}}}\\\\\\\boxed{\boxed{log_{b}(x\cdot y)=log_{b}(x)+log_{b}(y)}}\\\\\\\boxed{\boxed{log_{b}\left(\dfrac{x}{y}\right)=log_{b}(x)-log_{b}(y)}}\\\\\\\boxed{\boxed{log_{b}(a^{n})=n\cdot log_{b}(a)}}")
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![log_{3}\left(\dfrac{a^{2}\sqrt{b}}{\sqrt[3]{c}}\right)=log_{3}(a^{2})+log_{3}(\sqrt{b})-log_{3}\sqrt[3]{c}\\\\\\log_{3}\left(\dfrac{a^{2}\sqrt{b}}{\sqrt[3]{c}}\right)=2\cdot log_{3}(a)+log_{3}(b^{1/2})-log_{3}(c^{1/3})\\\\\\log_{3}\left(\dfrac{a^{2}\sqrt{b}}{\sqrt[3]{c}}\right)=2\cdot log_{3}(a)+\dfrac{1}{2}log_{3}(b)-\dfrac{1}{3}log_{3}(c)](https://tex.z-dn.net/?f=log_%7B3%7D%5Cleft%28%5Cdfrac%7Ba%5E%7B2%7D%5Csqrt%7Bb%7D%7D%7B%5Csqrt%5B3%5D%7Bc%7D%7D%5Cright%29%3Dlog_%7B3%7D%28a%5E%7B2%7D%29%2Blog_%7B3%7D%28%5Csqrt%7Bb%7D%29-log_%7B3%7D%5Csqrt%5B3%5D%7Bc%7D%5C%5C%5C%5C%5C%5Clog_%7B3%7D%5Cleft%28%5Cdfrac%7Ba%5E%7B2%7D%5Csqrt%7Bb%7D%7D%7B%5Csqrt%5B3%5D%7Bc%7D%7D%5Cright%29%3D2%5Ccdot+log_%7B3%7D%28a%29%2Blog_%7B3%7D%28b%5E%7B1%2F2%7D%29-log_%7B3%7D%28c%5E%7B1%2F3%7D%29%5C%5C%5C%5C%5C%5Clog_%7B3%7D%5Cleft%28%5Cdfrac%7Ba%5E%7B2%7D%5Csqrt%7Bb%7D%7D%7B%5Csqrt%5B3%5D%7Bc%7D%7D%5Cright%29%3D2%5Ccdot+log_%7B3%7D%28a%29%2B%5Cdfrac%7B1%7D%7B2%7Dlog_%7B3%7D%28b%29-%5Cdfrac%7B1%7D%7B3%7Dlog_%7B3%7D%28c%29 "log_{3}\left(\dfrac{a^{2}\sqrt{b}}{\sqrt[3]{c}}\right)=log_{3}(a^{2})+log_{3}(\sqrt{b})-log_{3}\sqrt[3]{c}\\\\\\log_{3}\left(\dfrac{a^{2}\sqrt{b}}{\sqrt[3]{c}}\right)=2\cdot log_{3}(a)+log_{3}(b^{1/2})-log_{3}(c^{1/3})\\\\\\log_{3}\left(\dfrac{a^{2}\sqrt{b}}{\sqrt[3]{c}}\right)=2\cdot log_{3}(a)+\dfrac{1}{2}log_{3}(b)-\dfrac{1}{3}log_{3}(c)")
Sustituindo os logaritmos por r, s e t:
![log_{3}\left(\dfrac{a^{2}\sqrt{b}}{\sqrt[3]{c}}\right)=2\cdot r+\dfrac{1}{2}\cdot s-\dfrac{1}{3}\cdot t\\\\\\log_{3}\left(\dfrac{a^{2}\sqrt{b}}{\sqrt[3]{c}}\right)=\dfrac{12r}{6}+\dfrac{3s}{6}-\dfrac{2t}{6}\\\\\\\boxed{\boxed{log_{3}\left(\dfrac{a^{2}\sqrt{b}}{\sqrt[3]{c}}\right)=\dfrac{1}{6}(12r+3s-2t)}}](https://tex.z-dn.net/?f=log_%7B3%7D%5Cleft%28%5Cdfrac%7Ba%5E%7B2%7D%5Csqrt%7Bb%7D%7D%7B%5Csqrt%5B3%5D%7Bc%7D%7D%5Cright%29%3D2%5Ccdot+r%2B%5Cdfrac%7B1%7D%7B2%7D%5Ccdot+s-%5Cdfrac%7B1%7D%7B3%7D%5Ccdot+t%5C%5C%5C%5C%5C%5Clog_%7B3%7D%5Cleft%28%5Cdfrac%7Ba%5E%7B2%7D%5Csqrt%7Bb%7D%7D%7B%5Csqrt%5B3%5D%7Bc%7D%7D%5Cright%29%3D%5Cdfrac%7B12r%7D%7B6%7D%2B%5Cdfrac%7B3s%7D%7B6%7D-%5Cdfrac%7B2t%7D%7B6%7D%5C%5C%5C%5C%5C%5C%5Cboxed%7B%5Cboxed%7Blog_%7B3%7D%5Cleft%28%5Cdfrac%7Ba%5E%7B2%7D%5Csqrt%7Bb%7D%7D%7B%5Csqrt%5B3%5D%7Bc%7D%7D%5Cright%29%3D%5Cdfrac%7B1%7D%7B6%7D%2812r%2B3s-2t%29%7D%7D "log_{3}\left(\dfrac{a^{2}\sqrt{b}}{\sqrt[3]{c}}\right)=2\cdot r+\dfrac{1}{2}\cdot s-\dfrac{1}{3}\cdot t\\\\\\log_{3}\left(\dfrac{a^{2}\sqrt{b}}{\sqrt[3]{c}}\right)=\dfrac{12r}{6}+\dfrac{3s}{6}-\dfrac{2t}{6}\\\\\\\boxed{\boxed{log_{3}\left(\dfrac{a^{2}\sqrt{b}}{\sqrt[3]{c}}\right)=\dfrac{1}{6}(12r+3s-2t)}}")
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Sustituindo os logaritmos por r, s e t:
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