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why is ln[log^0.9] undefined?

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I don't think the question is typed clearly enough ln(log(30)^(0.9))=0.35108558... so that is defined.
@TuringTest where did you get log(30) from?

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I'm just using an example log is not defined for x<=0, so unless that is the case for one of the arguments it is not a true statement.
logx not defined for x<=0 *
Right, my point is, is that the question does not taking the logarithm of anything: ln[(log( ))^0.9]
Oh, then I just didn't understand that you were saying that. I agree.
It is likely that the question was typed out wrong... who knows.
log^.9 means nothing, so it must be

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