Question

Simplify: log base 3 (1/3) ...?

Answer 1

The answer is 1.

$log_3( \frac{1}{3} )$
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Since $log_y(x) = \frac{log_{10}(x)}{log_{10}(y)}$, then: $log_3( \frac{1}{3} )= \frac{log_{10} ( \frac{1}{3}) }{log_{10}(3)}$
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Since $\frac{1}{x} = x^{-1}$, then: $\frac{log_{10} ( \frac{1}{3}) }{log_{10}(3)} = \frac{log_{10}(3^{-1} )}{log_{10}(3)}$
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Since $log x^{a} =a*logx$, then: $\frac{log_{10}(3^{-1} )}{log_{10}(3)}= \frac{-1*log_{10}(3)}{log_{10}(3)}$

From here: $\frac{-1*log_{10}(3)}{log_{10}(3)}=-1*\frac{log_{10}(3)}{log_{10}(3)}=-1*1 = -1$