To simplify and evaluate the expression

$\frac{\log\left(\frac{18000 \times \frac{0.05}{12}}{600} + 1\right)}{12 \log\left(1 + \frac{0.05}{12}\right)}$

let's break it down step by step.

Step 1: Calculate the numerator

First, calculate $\frac{18000 \times \frac{0.05}{12}}{600} + 1$.

1. Calculate $\frac{0.05}{12}$: $\frac{0.05}{12} \approx 0.00416667$
2. Multiply by 18000: $18000 \times 0.00416667 \approx 75$
3. Divide by 600: $\frac{75}{600} = 0.125$
4. Add 1: $0.125 + 1 = 1.125$

So, the numerator becomes: $\log(1.125)$

Step 2: Calculate the denominator

Next, calculate $12 \log\left(1 + \frac{0.05}{12}\right)$.

1. Calculate $1 + \frac{0.05}{12}$: $1 + 0.00416667 = 1.00416667$
2. Calculate $\log(1.00416667)$: $\log(1.00416667) \approx 0.001803$
3. Multiply by 12: $12 \times 0.001803 \approx 0.021636$

Step 3: Combine results

Now, substitute back into the expression: $\frac{\log(1.125)}{0.021636}$

Calculate $\log(1.125)$: $\log(1.125) \approx 0.053072$

Finally, divide: $\frac{0.053072}{0.021636} \approx 2.45$

Final Result

The value of the expression is approximately $2.45$.

Would you like more details or explanations about any part of the calculation? Here are some related questions you might find interesting:

1. How does logarithmic properties simplify calculations?
2. What is the significance of the parameters in the expression?
3. Can this expression be applied in real-world financial scenarios?
4. How does changing the interest rate affect the outcome?
5. What other mathematical tools can help in similar calculations?

Tip: When working with logarithms, remember to check if the values are within the domain of the logarithmic function to avoid errors.