To minimize $(x - y)^2 + (z - 2)^2$, let us carefully analyze and solve the problem.

Step 1: Simplify the logarithmic equation

Given: $\ln x + \ln 2 = \ln z$ Using the logarithmic property $\ln a + \ln b = \ln(ab)$, we simplify: $\ln(2x) = \ln z \implies z = 2x.$

Step 2: Relate $y$ and $z$ using the second equation

The second equation is: $yz = 4.$ Substituting $z = 2x$ into $yz = 4$, we have: $y(2x) = 4 \implies y = \frac{2}{x}.$

Step 3: Rewrite the objective function

The objective function to minimize is: $(x - y)^2 + (z - 2)^2.$ Substituting $y = \frac{2}{x}$ and $z = 2x$, it becomes: $\left(x - \frac{2}{x}\right)^2 + (2x - 2)^2.$

Step 4: Analyze the function

Define $f(x)$ as: $f(x) = \left(x - \frac{2}{x}\right)^2 + (2x - 2)^2.$

Expand the terms:

1. Expand the first term: $\left(x - \frac{2}{x}\right)^2 = x^2 - 2x \cdot \frac{2}{x} + \left(\frac{2}{x}\right)^2 = x^2 - 4 + \frac{4}{x^2}.$

2. Expand the second term: $(2x - 2)^2 = 4x^2 - 8x + 4.$

Thus, $f(x)$ becomes: $f(x) = \left(x^2 - 4 + \frac{4}{x^2}\right) + \left(4x^2 - 8x + 4\right).$ Combine terms: $f(x) = 5x^2 - 8x + \frac{4}{x^2}.$

Step 5: Minimize $f(x)$

To minimize $f(x)$, compute the derivative and set it to zero.

Compute $f'(x)$:

$f'(x) = 10x - 8 - \frac{8}{x^3}.$ Set $f'(x) = 0$: $10x - 8 - \frac{8}{x^3} = 0.$

Solve for $x$:

Multiply through by $x^3$ to eliminate the fraction: $10x^4 - 8x^3 - 8 = 0.$ Factorize: $2(5x^4 - 4x^3 - 4) = 0.$ We need to solve $5x^4 - 4x^3 - 4 = 0$. Numerical methods or approximations are necessary here.

Step 6: Numerical approximation

Using numerical solvers, we find the critical value of $x$ that minimizes $f(x)$. Plug this value back into $y = \frac{2}{x}$ and $z = 2x$ to compute the minimum value of the objective function.

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