I know that log of a negative number is not possible but, $\log(-5)^2$ is possible. Therefore $\log(-5)^2=2\log(-5)$ but $\log(-5)$ is not possible but $log$ of $-5$ square is possible ....can anyone explain this? Thanks

  • $\begingroup$ Hello. Why do you say that log(-5)^2 is possible? $\endgroup$ – Mankind Jun 10 '15 at 11:21
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    $\begingroup$ Maybe you are confused about the difference between $log((-5)^2)$ and $log(-5)^2$?, i guess you are asking about why the first expression doesn't allow us to define the logarithm for negative numbers. the boring reason is just that the rule you are using doesn't work for negative numbers. $\endgroup$ – An.Ditlev Jun 10 '15 at 11:22
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    $\begingroup$ You can always save this messes doing $\;\log(x^n)=n\log|x|\;$ , if you allow negative values of $\;x\;$ and even powers of it. $\endgroup$ – Timbuc Jun 10 '15 at 11:35

There is no reason to expect that $$ \log a^b = b\log a $$ holds for $a<0$.

(Once you get to complex logarithms, you can make sense of such equalities, but only if you allow multi-valued interpretations of the logarithm.)


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