I think you're making a number of erroneous assumptions. You have $f(x)=P(X=x)=c/x$ is the probability distribution of $X$ and the state space is $1,3,\ldots,3^m$. It seems like the problem is asking you to find the normalization $c$ for the probability distribution. To wit, you must have that:
$1=\sum_{i=0}^m P(X=3^i)=\sum_{i=0}^m \frac{c}{3^i}=c\frac{1-(1/3)^{m+1}}{1-(1/3)}$.
which you can check by a similar argument to Peter's answer. Now solve for $c$.
The point is that you don't want to say that $P(X=x)=c/x=1$ for every $x$, that doesn't make much sense since you'd be assigning probability 1 to each event. It's the sum that should equal 1. As well, take care in interpreting $P(X=x)$, this is lingo for asking the probability of the random variable $X$ being equal to $x$, which the problem says is proportional to $1/x$. The point is $x$ lives in the range of $X$, so it doesn't make much sense to try and solve for $x$.