$\frac{\ln(x^2)}{\ln(x)} = 2$?
Upon trying to evaluate $\frac{\ln(x^2)}{\ln(x)}$, i've found that google plots it as always equal to 2, other than 0 where it is undefined. Why is this the case?
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Recall the logarithm property $$\frac{\ln x^2}{\ln x} = \frac{2\ln x}{\ln x} = 2.$$ But this is only true when $x>0$ and $x\neq1$. Otherwise, there is a "hole" there; a removable discontinuity. Notice that this is difficult not to graph, so graphing tools usually just fill the hole/graph over it.
Em.
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but if $x\leq0$ $\ln{x}$ is undefined, why have a removable discontinuity? – AsdrubalBeltran Feb 05 '16 at 02:43
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@AsdrubalBeltran ............ at $x = 1$. – Em. Feb 05 '16 at 02:44
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@AsdrubalBeltran There is no plot if $x\leq 0$. – Em. Feb 05 '16 at 02:45
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ok yes thanks, is in $x=1$ – AsdrubalBeltran Feb 05 '16 at 02:46
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@AsdrubalBeltran Ok, glad you understand. – Em. Feb 05 '16 at 02:47
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We can use the change of base formula to write the following. $$\frac{\ln \left(x^2\right)}{\ln x}=\log_x\!\!\left(x^2\right)=2$$ The above holds for all $x>0,x\neq1$.
Akiva Weinberger
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Lythia
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