How to solve something like this?
$\displaystyle |3x-2|^{\sqrt{2x-3x^2}}=1$
$\displaystyle 2x-3x^2=0$ doesn't work, because here x=2/3 and x=0, and the answer is 1/3...
The exponent will be 0 OR the base will be $\displaystyle \pm 1$.
$\displaystyle 2x-3x^2 = 0$ which is $\displaystyle x=0$. x cannot be 2/3 because it would give the undefined $\displaystyle 0^0$
You can also solve $\displaystyle |3x-2| = 1$ - one of the solutions is 1/3
edit: Not sure why 0 cannot be a solution because $\displaystyle f(0) = |3(0)-2|^{\sqrt{0-0}} = (2)^0 = 1$
The reason $\displaystyle x=2/3$ isn't working is that this gives you $\displaystyle 0^0=1$, 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:
$\displaystyle |3x-2|^{\sqrt{2x-3x^2}}=1$
Split it into:
1) $\displaystyle (3x-2)^{\sqrt{2x-3x^2}}=1$
and
2) $\displaystyle (2-3x)^{\sqrt{2x-3x^2}}=1$
Take logs of both, starting with:
1) $\displaystyle (3x-2)^{\sqrt{2x-3x^2}}=1$
$\displaystyle \sqrt{2x-3x^2}\log(3x-2)=0$
Either $\displaystyle 3x-2 = 1$ (this gives $\displaystyle log(1) = 0$)
In which case $\displaystyle x = 1$
Or as we've already established, $\displaystyle \sqrt{2x-3x^2}=0$
However neither work! The $\displaystyle x=1$ solution, when resubstituted, gives a complex value and is thus omitted. We've already gathered that the $\displaystyle \sqrt{2x-3x^2}=0$ doesn't work either. So then, we have to look at
2) $\displaystyle (2-3x)^{\sqrt{2x-3x^2}}=1$
Taking logs again:
$\displaystyle \sqrt{2x-3x^2}\log(2-3x)=Log(1)$
The only untested possibility is $\displaystyle 2-3x=1$
In which case, $\displaystyle -1=-3x$
$\displaystyle x=\frac{1}{3}$
Interestingly, with my approach, the $\displaystyle x=0$ 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.
Quacky, I'm not sure what is the base for you logarithms? Do you mean $\displaystyle lg$?
I also don't understand why x can't be = 0. Because it is included in the domain of the equation. $\displaystyle x\in[0;\frac{2}{3}]$
And as e^(i*pi) wrote $\displaystyle f(0) = |3(0)-2|^{\sqrt{0-0}} = (2)^0 = 1$
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.