Solve Complex Logarithm Equation: Product of Log_a(x), Log_b(y), and Log_c(2)

To solve this problem, we must examine both sides of the equation:

The left-hand side of the equation:
$\log_a x \log_b y \log_c 2$

The right-hand side of the equation:
$(\log_a y^3 - \log_a y^2)(\log_b \frac{1}{2} + \log_b 2^2)\log_c(x^2+1)$

Let's simplify and understand both sides:

• For $\log_a y^3 - \log_a y^2$, apply the power rule of logarithms:
  $\log_a y^3 = 3 \log_a y$ and $\log_a y^2 = 2 \log_a y$
  Thus, $\log_a y^3 - \log_a y^2 = (3 \log_a y - 2 \log_a y) = \log_a y$.
• For $\log_b \frac{1}{2} + \log_b 2^2$, apply the rules:
  $\log_b \frac{1}{2} = \log_b 1 - \log_b 2 = -\log_b 2$ and $\log_b 2^2 = 2 \log_b 2$
  Thus, $\log_b \frac{1}{2} + \log_b 2^2 = (-\log_b 2 + 2 \log_b 2) = \log_b 2$.
• Combine the simplifications for the right side:
  $\log_a y \cdot \log_b 2 \cdot \log_c (x^2 + 1)$.

Now the equation simplifies to:
$\log_a x \log_b y \log_c 2 = \log_a y \log_b 2 \log_c (x^2 + 1)$

By inspection:

• Both sides involve products of terms with different bases, which complicates direct comparison except where specific values are chosen.
• Due to the nature of logarithms, for equalities of this form, the left-hand side and right-hand side must somehow equate if solutions exist.
• Upon trying specific values became apparent as non-simple iterations seem to contradict basic logarithmic properties, ultimately showing complexities in natural number solutions.

Under these stringent conditions, it leads us to conclude:

Therefore, the solution to the given problem is No solution.