<s>[INST] If $\log _b a\cdot\log_c a +\log _a b\cdot\log_c b+\log _a c\cdot\log_b c=3$ then find the value of $abc$ 
If $\log _b a\cdot\log_c a +\log _a b\cdot\log_c b+\log _a c\cdot\log_b c=3$ and $a,b,c$ are different positive real numbers not equal to 1, then find the value of $abc$.    

I tried to simplify this by different methods like using the identity $\log_b a=\frac{1}{\log_a b}$, but I couldn't get anywhere. Addition, subraction can't be used. I am not able to figure out a way to simplify the LHS. It would be great if someone could help me to proceed with this problem.
 [/INST] If you set $X=\log a$, $Y=\log b$ and $Z=\log c$ (any base you like), your condition becomes easily
$$
\frac{X^3+Y^3+Z^3}{XYZ}=3
$$
Now you can use the identity
$$
X^3+Y^3+Z^3=(X+Y+Z)^3-3(X+Y+Z)(XY+YZ+ZX)+3XYZ
$$
to get that
$$
(X+Y+Z)\bigl((X+Y+Z)^2-3(XY+YZ+ZX)\bigr)=0
$$
The second factor can be written
$$
X^2+Y^2+Z^2-XY-YZ-ZX
$$
and we should consider the quadratic form having as matrix
$$
A=\begin{bmatrix}
1 & -1/2 & -1/2 \\
-1/2 & 1 & -1/2 \\
-1/2 & -1/2 & 1
\end{bmatrix}
$$
Since
$$
\det[1]=1>0,
\qquad
\det\begin{bmatrix}1 & -1/2 \\ -1/2 & 1\end{bmatrix}=3/4>0
\qquad
\det A=0
$$
the quadratic form is positive semidefinite. Its null space contains the vector $[1\;1\;1]^T$, so we can conclude that your condition implies
$$
X+Y+Z=0\qquad\text{or}\qquad X=Y=Z
$$
Since by assumption $a$, $b$ and $c$ are pairwise distinct, we end with $X+Y+Z=0$.

Without quadratic forms, you can reason about $X^2+Y^2+Z^2-XY-YZ-ZX=0$ as follows. Since $Z\ne0$ by assumption, we can set $X=uZ$ and $Y=vZ$, so the equation becomes
$$
u^2-uv+v^2-u-v+1=0
$$
and, solving with respect to $v$, $v^2-v(u+1)+u^2-u+1=0$, the discriminant is
$$
(u+1)^2-4(u^2-u+1)=-3(u-1)^2
$$
so the equation has a solution only for $u=1$, which gives $v=1$. Therefore $X=Y=Z$.

Another “elementary” approach. Suppose $(X+Y+Z)^2=3(XY+YZ+ZX)$. Set $s=X+Y+Z$ and $p=XYZ$; then $X$, $Y$ and $Z$ are the roots of the equation
$$
t^3-st^2+\frac{s^2}{3}t-p=0
$$
by Viète's formulas. We can complete the cube getting
$$
\left(t-\frac{s}{3}\right)^3=p-\frac{s^3}{27}
$$
which should have three real roots. This is impossible unless the roots are coincident, so $p=s^3/27$ and $X=Y=Z$.
</s>