The exponent will be 0 OR the base will be .
which is . x cannot be 2/3 because it would give the undefined
You can also solve - one of the solutions is 1/3
edit: Not sure why 0 cannot be a solution because
The reason isn't working is that this gives you , which I don't think is defined. Anyway, here is my very poor and untidy method, which does give the right answer in a 'round-the-houses' approach:
Split it into:
1)
and
2)
Take logs of both, starting with:
1)
Either (this gives )
In which case
Or as we've already established,
However neither work! The solution, when resubstituted, gives a complex value and is thus omitted. We've already gathered that the doesn't work either. So then, we have to look at
2)
Taking logs again:
The only untested possibility is
In which case,
Interestingly, with my approach, the root indeed doesn't work - it gives negative values inside the logarithms, which are undefined, but that wasn't stated initially by the question. I assume whoever made the question took a similar approach as I did, or did something else, in which the 0 solution was lost along the way.
I know my method isn't sound, but it does explain the textbook's answer.
Yes, that's what I mean, although the base of the log doesn't matter in this case, as long as it's consistent.
I'm not sure either, as I said above, I supposed that someone followed a similar method as I did in which the 0 solution was accidentally dropped. Perhaps I didn't communicate my point very well.