Question 196625
*[Tex \LARGE \log_{8}\left(\sqrt{80}\right)-\log_{8}\left(\sqrt{5}\right)] ... Start with the given expression.



*[Tex \LARGE \log_{8}\left(\frac{\sqrt{80}}{\sqrt{5}}\right)] ... Combine the logs using the identity {{{log(b,(A))-log(b,(B))=log(b,(A/B))}}}.



*[Tex \LARGE \log_{8}\left(\frac{\sqrt{16\ast5}}{\sqrt{5}}\right)] ... Factor 80 into 16*5.



*[Tex \LARGE \log_{8}\left(\frac{\sqrt{16}\sqrt{5}}{\sqrt{5}}\right)] ... Break up the square root.



*[Tex \LARGE \log_{8}\left(\frac{4\sqrt{5}}{\sqrt{5}}\right)] ... Take the square root of 16 to get 4.



*[Tex \LARGE \log_{8}\left(4\right)] ... Reduce



*[Tex \LARGE \frac{\log_{10}\left(4\right)}{\log_{10}\left(8\right)}] ... Use the change of base formula




Remember the change of base formula is {{{log(b,(x))=log(10,(x))/log(10,(b))}}}




*[Tex \LARGE \frac{\log_{10}\left(2^2\right)}{\log_{10}\left(2^3\right)}] ... Rewrite each argument as a power of 2



*[Tex \LARGE \frac{2\log_{10}\left(2\right)}{3\log_{10}\left(2\right)}] ... Pull down the exponents using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}



*[Tex \LARGE \frac{2}{3}] ... Cancel out the common terms.




So *[Tex \LARGE \log_{8}\left(\sqrt{80}\right)-\log_{8}\left(\sqrt{5}\right)=\frac{2}{3}]