Question 191346
I'll do the first two to get you started


# 1


*[Tex \LARGE \log_{9}\left(\frac{1}{3}\right)] ... Start with the given expression.



*[Tex \LARGE \frac{\log_{10}\left(\frac{1}{3}\right)}{\log_{10}(9)}] ... Use the change of base formula



*[Tex \LARGE \frac{\log_{10}\left(3^{-1}\right)}{\log_{10}(3^2)}] ... Rewrite each term as an exponential expression with a base of 3



*[Tex \LARGE \frac{-\log_{10}\left(3\right)}{2\log_{10}(3)}] ... Pull down the exponents.



*[Tex \LARGE -\frac{1}{2}] ... Cancel out the common terms.



So *[Tex \LARGE \log_{9}\left(\frac{1}{3}\right)=-\frac{1}{2}]



<hr>

*[Tex \LARGE \log_{\frac{1}{4}}\left(16\right)] ... Start with the given expression.



*[Tex \LARGE \frac{\log_{10}(16)}{\log_{10}\left(\frac{1}{4}\right)}]  ... Use the change of base formula



*[Tex \LARGE \frac{\log_{10}(4^2)}{\log_{10}\left(4^{-1}\right)}] ... Rewrite each term as an exponential expression with a base of 4



*[Tex \LARGE \frac{2\log_{10}(4)}{-\log_{10}\left(4\right)}] ... Pull down the exponents.




*[Tex \LARGE \frac{2}{-1}] ... Cancel out the common terms.



*[Tex \LARGE -2] ... Reduce



So *[Tex \LARGE \log_{\frac{1}{4}}\left(16\right)=-2]




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Edit: Here's the last solution:


# 3



*[Tex \LARGE \log_{8}\left(\frac{1}{256}\right)] ... Start with the given expression.


*[Tex \LARGE \frac{\log_{10}\left(\frac{1}{256}\right)}{\log_{10}\left(8\right)}] ... Use the change of base formula



*[Tex \LARGE \frac{\log_{10}\left(2^{-8}\right)}{\log_{10}\left(2^{3}\right)}] ... Rewrite each term as an exponential expression with a base of 2



*[Tex \LARGE \frac{-8\log_{10}\left(2\right)}{3\log_{10}\left(2\right)}] ... Pull down the exponents.



*[Tex \LARGE \frac{-8}{3}] ... Cancel out the common terms.



So *[Tex \LARGE \log_{8}\left(\frac{1}{256}\right)=-\frac{8}{3}]