We are tasked with finding the minimum value of $(x-y)^2 + (z-2)^2$ under the given constraints:

1. $\ln x + \ln 2 = \ln z$
2. $yz = 4$

Step-by-Step Solution:

1. Simplify the logarithmic equation

From $\ln x + \ln 2 = \ln z$, we use the logarithmic property $\ln a + \ln b = \ln(ab)$: $\ln(2x) = \ln z$ Thus, $z = 2x$.

2. Substitute $z = 2x$ into $yz = 4$

Since $z = 2x$, we substitute into the second constraint $yz = 4$: $y(2x) = 4 \quad \implies \quad yx = 2$

3. Express $y$ in terms of $x$

From $yx = 2$, we get: $y = \frac{2}{x}$

4. Write the objective function in terms of $x$

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

5. Expand and simplify the function

Expand $f(x)$: $f(x) = \left(x - \frac{2}{x}\right)^2 + (2x - 2)^2$ $f(x) = \left(x^2 - 2 + \frac{4}{x^2}\right) + \left(4x^2 - 8x + 4\right)$ $f(x) = x^2 - 2 + \frac{4}{x^2} + 4x^2 - 8x + 4$ $f(x) = 5x^2 - 8x + \frac{4}{x^2} + 2$

6. Minimize $f(x)$

To find the minimum, compute the derivative of $f(x)$ with respect to $x$: $f'(x) = 10x - 8 - \frac{8}{x^3}$ Set $f'(x) = 0$: $10x - 8 - \frac{8}{x^3} = 0$ $10x^4 - 8x^3 - 8 = 0$ Simplify: $2x^4 - x^3 - 1 = 0$

7. Solve for $x$

The equation $2x^4 - x^3 - 1 = 0$ requires numerical or approximate methods to solve. Using approximation, $x \approx 1.17$ satisfies this equation.

8. Compute $y$ and $z$

Using $x \approx 1.17$: $y = \frac{2}{x} \approx \frac{2}{1.17} \approx 1.71$ $z = 2x \approx 2(1.17) \approx 2.34$

9. Compute the objective function

Substitute $x \approx 1.17$, $y \approx 1.71$, and $z \approx 2.34$ into $f(x)$: $f(x) = (x-y)^2 + (z-2)^2$ $f(x) \approx (1.17 - 1.71)^2 + (2.34 - 2)^2$ $f(x) \approx (-0.54)^2 + (0.34)^2$ $f(x) \approx 0.2916 + 0.1156 \approx 0.4072$

Final Answer:

The minimum value of $(x-y)^2 + (z-2)^2$ is approximately: $\boxed{0.41}$

Would you like further details on solving this, or have any related questions?

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Related Questions:

1. How do logarithmic properties simplify equations in optimization problems?
2. Can the solution to $2x^4 - x^3 - 1 = 0$ be found exactly, and how?
3. How does $f(x) = 5x^2 - 8x + \frac{4}{x^2} + 2$ behave for small or large $x$?
4. What numerical methods are best for solving quartic equations like $2x^4 - x^3 - 1 = 0$?
5. How does the derivative $f'(x)$ provide insights into critical points?

Tip:

When minimizing a function with constraints, always substitute constraints early to simplify the objective function.