The answer is 1.
[tex]log_3( \frac{1}{3} )[/tex]
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Since [tex]log_y(x) = \frac{log_{10}(x)}{log_{10}(y)} [/tex], then: [tex]log_3( \frac{1}{3} )= \frac{log_{10} ( \frac{1}{3}) }{log_{10}(3)} [/tex]
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Since [tex] \frac{1}{x} = x^{-1} [/tex], then: [tex]\frac{log_{10} ( \frac{1}{3}) }{log_{10}(3)} = \frac{log_{10}(3^{-1} )}{log_{10}(3)} [/tex]
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Since [tex]log x^{a} =a*logx[/tex], then: [tex]\frac{log_{10}(3^{-1} )}{log_{10}(3)}= \frac{-1*log_{10}(3)}{log_{10}(3)} [/tex]
From here: [tex]\frac{-1*log_{10}(3)}{log_{10}(3)}=-1*\frac{log_{10}(3)}{log_{10}(3)}=-1*1 = -1[/tex]
