Solve the Logarithmic Equation: log₄(x²)·log₇16 = 2log₇8

To solve this logarithmic equation, we will break down and simplify the given expression step by step:

Step 1: Simplify each logarithm using the change of base formula.

First, consider $\log_4 x^2$:
Using the power rule, $\log_4 x^2 = 2 \log_4 x$.
Now apply the change of base formula:
$\log_4 x = \frac{\log x}{\log 4}$, thus $\log_4 x^2 = 2 \cdot \frac{\log x}{\log 4}$.

Step 2: Simplify $\log_7 16$ and $\log_7 8$ using the change of base formula.
$\log_7 16 = \frac{\log 16}{\log 7} = \frac{\log (2^4)}{\log 7} = \frac{4 \log 2}{\log 7}$.
Similarly, $\log_7 8 = \frac{\log 8}{\log 7} = \frac{\log (2^3)}{\log 7} = \frac{3 \log 2}{\log 7}$.

Step 3: Substitute these values back into the equation.
$\frac{2 \log x}{\log 4} \cdot \frac{4 \log 2}{\log 7} = 2 \cdot \frac{3 \log 2}{\log 7}$

Step 4: Simplify the equation by canceling out common terms and solving for $\log x$.
After cancelling $\frac{\log 2}{\log 7}$ from both sides, we have:
$\frac{8 \log x}{\log 4} = 6$.

Step 5: Calculate $\log 4 = 2 \log 2$, so substitute:
$\frac{8 \log x}{2 \log 2} = 6 \implies 4 \log x = 6 \log 2$, thus $\log x = \frac{3}{2} \log 2$.

Step 6: Solve for $x$ using exponentiation.
Since $\log x = \frac{3}{2} \log 2$, exponentiation gives $x = 2^{\frac{3}{2}} = \sqrt{8}$. However, since logarithms are defined for positive numbers, we must consider $\pm$ for solutions within the constraints. Thus, $x = \pm \sqrt{8}$.

Therefore, the solution to the problem is $x = \pm\sqrt{8}$, corresponding to choice $4$.