Question

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

Answer 1

The answer is 1.


`log_3( (1)/(3) )`
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Since
`log_y(x) = (log_(10)(x))/(log_(10)(y))`, then:
`log_3( (1)/(3) )= (log_(10) ( (1)/(3)) )/(log_(10)(3))`
_____
Since
`(1)/(x) = x^(-1)`, then:
`(log_(10) ( (1)/(3)) )/(log_(10)(3)) = (log_(10)(3^(-1) ))/(log_(10)(3))`
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Since
`log x^(a) =a*logx`, then:
`(log_(10)(3^(-1) ))/(log_(10)(3))= (-1*log_(10)(3))/(log_(10)(3))`

From here:
`(-1*log_(10)(3))/(log_(10)(3))=-1*(log_(10)(3))/(log_(10)(3))=-1*1 = -1`