What is the distribution of real numbers with biased digits? Suppose I have an infinite sequence of biased bits where the probability of $1$ is $2/3$ and the probability of $0$ is $1/3.$ If I view these as the digits in the binary expansion of a real number, then this sequence defines a real number in the interval $[0,1]$. So what kind of distribution does this real number have?
Some considerations I have made so far is that the probability between $0.5$ and $1$ should be twice the probability between $0$ and $0.5.$ Similarly the probability between $0.25$ and $0.5$ should be twice the probability between $0$ and $0.25.$ A general way of writing this is recursive relationship is
$$F(2x) - F(x) = 2F(x).$$
Adding boundary conditions I get the three equations
$$F(0)=0\\
F(1)=1\\
F(2x)=3F(x)$$
which, if viewed as a recurrence relation, has the solution $F(x) = x^{\log_2(3)}$. My question is: Is this really airtight? Setting up these equations and using the solution from a recurrence relation felt a little hand wavy. I can easily verify that $x^{\log_2(3)}$ satisfies the above conditions for real numbers in the interval $[0,1]$, but is this solution unique?

Suppose for $0\le x\le 1$ we have $F(x) = \Pr(X\le x) = x^{\log_2 3}.$
Then $F(0) = 0$ and $F(1/2) = 1/3$ and $F(1)=1,$ all as expected.
Let $D_1,D_2,D_3,\ldots$ be the binary digits of $X.$ We wanted these to be i.i.d. with each equal to $1$ with probability $2/3.$
Observe that this means $\displaystyle X = \sum_{k=1}^\infty \frac{D_k}{2^k}$ and $\displaystyle 2(X - D_1/2) = 2X-D_1 = \sum_{k=2}^\infty \frac{D_k}{2^k} $ both have the same distribution, since the two sequences $(D_1,D_2,D_3,\ldots)\vphantom{\dfrac11}$ and $(D_2,D_3,D_3,\ldots)$ both have the same distribution. The conditional distribution of $2X-D_1$ given $D_1$ has this same distribution, and since $2X-D_1$ is determined by $D_2,D_3,D_4,\ldots,$ we have $2X-D_1$ independent of $D_1.$ Since $2X-D_1$ is independent of $D_1$ and $2X-D_1$ has the same distribution that $X$ has, we can say
\begin{align}
F(x) & = \Pr(X\le x) = \Pr(2X-D_1\le x) = \Pr(2X-D_1\le x \mid D_1=1) \\[10pt]
& = \frac{\Pr(2X-D_1 \le x\ \&\ D_1=1)}{\Pr(D_1=1)} = \frac{\Pr(1/2 \le X \le \frac{x+1} 2)}{2/3} = \frac{\left( \frac{x+1} 2 \right)^{\log_2 3} - 1/3}{2/3}.
\end{align}
Is the following true?
$$
x^{\log_2 3} \, \overset{\Large\text{?}} = \frac{\left( \frac{x+1} 2 \right)^{\log_2 3} - 1/3}{2/3}
$$
But some numerical computation gives counterexamples to this. They are not equal when $x=0.2.$
