To solve for the variables in the given equation:

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

We can approach this in multiple ways depending on the goal, such as maximizing or minimizing $L$ with respect to $X$, $Y$, and $\lambda$, or solving for a specific variable. This equation involves a Lagrange multiplier $\lambda$, which suggests we are dealing with an optimization problem subject to the constraint $5000 - 4X - 2Y = 0$.

Steps for finding the maximum or minimum:

1. Formulate the Lagrange equations: To solve this, we differentiate $L$ with respect to $X$, $Y$, and $\lambda$, and set the derivatives equal to zero (first-order necessary conditions for optimality).

   • Differentiate $L$ with respect to $X$: $\frac{\partial L}{\partial X} = \frac{0.8}{X \ln 10} - 4\lambda = 0$

   • Differentiate $L$ with respect to $Y$: $\frac{\partial L}{\partial Y} = \frac{0.2}{Y \ln 10} - 2\lambda = 0$

   • Differentiate $L$ with respect to $\lambda$: $\frac{\partial L}{\partial \lambda} = 5000 - 4X - 2Y = 0$

2. Solve the system of equations:

   • From the first equation: $\lambda = \frac{0.8}{4X \ln 10}$
   • From the second equation: $\lambda = \frac{0.2}{2Y \ln 10}$
   • Set the two expressions for $\lambda$ equal: $\frac{0.8}{4X \ln 10} = \frac{0.2}{2Y \ln 10}$ Simplify: $\frac{0.8}{4X} = \frac{0.2}{2Y} \quad \Rightarrow \quad \frac{1}{X} = \frac{1}{Y} \quad \Rightarrow \quad X = Y$

3. Substitute $X = Y$ into the constraint: Use the constraint equation: $5000 - 4X - 2Y = 0$ Substituting $X = Y$: $5000 - 4X - 2X = 0 \quad \Rightarrow \quad 5000 = 6X \quad \Rightarrow \quad X = \frac{5000}{6} \approx 833.33$ Since $X = Y$, $Y = 833.33$.

Thus, the solution is: $X = Y = 833.33$

Summary:

The optimal values for $X$ and $Y$ that satisfy the equation are approximately $833.33$.

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Would you like further details or have any questions on this? Here are some related questions you might find helpful:

1. How does the Lagrange multiplier method work in optimization problems?
2. Can we solve for $X$ and $Y$ if different constraints are applied?
3. What happens if we change the logarithmic coefficients (0.8 and 0.2)?
4. How would the solution change if $\lambda$ had a specific value?
5. How does the log function affect the behavior of the solution?

Tip: Always check the constraint equation after solving, as it helps confirm if the derived values satisfy the original problem condition.