To solve this function or optimize it, we will need to address it step-by-step. Let's break down the given function:

$L = 0.8 \log X + 0.2 \log Y + \lambda (5000 - 4X - 2Y)$

where $X$, $Y$, and $\lambda$ (likely a Lagrange multiplier) are variables.

Step 1: Lagrange Multiplier

The given expression looks like it involves the method of Lagrange multipliers for optimization under a constraint. The constraint seems to be $5000 - 4X - 2Y = 0$, based on the $\lambda$ term.

If you want to maximize or minimize $L$, you'll need to:

Step 2: Partial Derivatives

To apply the method of Lagrange multipliers, you should take the partial derivatives of $L$ with respect to $X$, $Y$, and $\lambda$, and set them equal to zero to solve for the critical points.

1. Partial derivative with respect to $X$:

$\frac{\partial L}{\partial X} = \frac{0.8}{X} - 4\lambda = 0$

2. Partial derivative with respect to $Y$:

$\frac{\partial L}{\partial Y} = \frac{0.2}{Y} - 2\lambda = 0$

3. Partial derivative with respect to $\lambda$:

$\frac{\partial L}{\partial \lambda} = 5000 - 4X - 2Y = 0$

Step 3: Solving the System of Equations

You now have a system of three equations:

1. $\frac{0.8}{X} = 4\lambda$
2. $\frac{0.2}{Y} = 2\lambda$
3. $5000 - 4X - 2Y = 0$

We can solve these equations to find $X$, $Y$, and $\lambda$.

From Equation (1):

$\lambda = \frac{0.8}{4X} = \frac{0.2}{Y}$ This gives the relationship $Y = 4X$.

Substituting $Y = 4X$ into Equation (3):

$5000 - 4X - 2(4X) = 0$ $5000 - 12X = 0$ $X = \frac{5000}{12} = 416.67$

Substituting $X = 416.67$ into $Y = 4X$:

$Y = 4(416.67) = 1666.67$

Step 4: Conclusion

The optimal values of $X$ and $Y$ are approximately: $X \approx 416.67, \quad Y \approx 1666.67$ You can find $\lambda$ from these values by substituting into one of the earlier equations.

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Would you like further clarification on any step? Here are 5 related questions to extend this topic:

1. How do we apply the method of Lagrange multipliers to other optimization problems?
2. What is the significance of logarithmic functions in optimization problems?
3. How do you interpret $\lambda$ in this type of optimization problem?
4. Could we solve this numerically if exact solutions weren't possible?
5. How do logarithmic derivatives help in economics or growth models?

Tip: Always check the constraints carefully in optimization problems to ensure you're not missing any additional limitations.