3. If $a, b, c$ form a geometric sequence, $\log _{c} a, \log _{b} c, \log _{a} b$ form an arithmetic sequence, then the common difference of this arithmetic sequence is $\qquad$ .
3.0 or $\frac{3}{2}$.

From $a 、 b 、 c$ forming a geometric sequence $\Rightarrow b^{2}=a c$
$$
\Rightarrow \log _{b} a+\log _{b} c=2 \text {. }
$$

Let $x=\log _{b} a, y=\log _{b} c$. Then $x+y=2$.
From $\log _{c} a 、 \log _{b} c 、 \log _{a} b$ forming an arithmetic sequence
$$
\begin{array}{l}
\Rightarrow \log _{c} a+\log _{a} b=2 \log _{b} c \\
\Rightarrow \frac{x}{y}+\frac{1}{x}=2 y .
\end{array}
$$

Then $2 x^{3}-9 x^{2}+9 x-2=0$
$$
\begin{array}{l}
\Rightarrow(x-1)\left(2 x^{2}-7 x+2\right)=0 \\
\Rightarrow x=1 \text { or } x=\frac{7 \pm \sqrt{33}}{4} .
\end{array}
$$

Therefore, $d=\frac{3}{2}$ or 0.