To find the minimum value of $(x-y)^2 + (z-2)^2$, given the equations:

1. $\ln x + \ln 2 = \ln(2x)$
2. $yz = 4$

We proceed as follows:

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Step 1: Rewrite the equations for clarity

• $\ln x + \ln 2 = \ln(2x)$ implies $x > 0$.
• $yz = 4$ implies $y > 0$ and $z > 0$.

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Step 2: Define the expression to minimize

The expression to minimize is: $f(x, y, z) = (x-y)^2 + (z-2)^2$

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Step 3: Solve for $z$ in terms of $y$

From $yz = 4$, we have: $z = \frac{4}{y}$

Substituting $z = \frac{4}{y}$ into $f(x, y, z)$: $f(x, y, z) = (x-y)^2 + \left(\frac{4}{y} - 2\right)^2$

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Step 4: Minimize the expression

1. Substitute $x = \frac{1}{2}$ (since $2x > 0$ for the logarithmic term):

   • From $\ln(2x)$, we know $x = \frac{1}{2}$.

2. Plug $x = \frac{1}{2}$ into $f(x, y, z)$: $f\left(\frac{1}{2}, y, z\right) = \left(\frac{1}{2} - y\right)^2 + \left(\frac{4}{y} - 2\right)^2$

3. Simplify the expression: $f\left(\frac{1}{2}, y\right) = \left(\frac{1}{2} - y\right)^2 + \left(\frac{4}{y} - 2\right)^2$

4. Minimize $f$ with respect to $y$: Take the derivative of $f$ with respect to $y$ and set it to 0: $f'(y) = 2\left(\frac{1}{2} - y\right)(-1) + 2\left(\frac{4}{y} - 2\right)\left(-\frac{4}{y^2}\right)$ Simplify the derivative: $f'(y) = -2\left(\frac{1}{2} - y\right) - \frac{8}{y^3}\left(\frac{4}{y} - 2\right)$

   Setting $f'(y) = 0$, solve for $y$ numerically or symbolically.

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Step 5: Verify the minimum

Once $y$ is found, substitute back into $z = \frac{4}{y}$ and $x = \frac{1}{2}$ to compute $(x-y)^2 + (z-2)^2$.

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Conclusion

The exact values depend on solving the derivative numerically or symbolically. The process minimizes $(x-y)^2 + (z-2)^2$ systematically.

Would you like a numerical solution or further explanation?

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Questions to Explore Further:

1. How is the derivative used in optimization problems?
2. What are the conditions for a global minimum in such problems?
3. Why does $x = \frac{1}{2}$ satisfy $\ln(2x)$?
4. How can symmetry help in minimizing quadratic expressions?
5. What tools can numerically solve non-linear equations?

Tip: For logarithmic equations, remember that combining terms using properties like $\ln(ab) = \ln a + \ln b$ can simplify the problem significantly.