Question 185729
{{{log (3, (-1))}}} Start with the given expression 
 


{{{y=log (3, (-1))}}} Set "y" equal to the expression



{{{3^y=-1}}} Use the property *[Tex \LARGE y=\log_{b}\left(x\right) \Leftrightarrow b^y=x] to rewrite the equation.




Now the question is: what value of "y" will make {{{3^y}}} equal to -1? Since there is NO value of "y" that will make {{{3^y}}} negative (because {{{3^y}}} is ALWAYS positive), this means that there are no solutions to {{{3^y=-1}}}



This in turn means that you CANNOT take the log of a negative number (or zero). 



So that's why {{{log (3, (-1))}}} is undefined



In fact, {{{log (b, (x))}}} is undefined where {{{b>1}}} and {{{x<=0}}} 



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{{{log (1, (1))}}} Start with the given expression



{{{log (10, (1))/log (10, (1))}}} Use the change of base formula {{{log(b,(x))=log(10,(x))/log(10,(b))}}} to rewrite the expression.



{{{0/0}}} Evaluate the log of 1 to get 0.



Now remember, division by zero is undefined. So we cannot go further.



So this shows us that {{{log (1, (1))}}} is undefined.