The question in the image asks, "3^50, 4^40, 5^30, और 6^20 में सबसे बड़ी संख्या कौन-सी है?" which translates to:

"Among $3^{50}, 4^{40}, 5^{30},$ and $6^{20}$, which is the largest number?"

We need to compare these numbers to find the largest.

Approach:

1. Take the natural logarithm (ln) of each expression to simplify comparison.

   • $\ln(3^{50}) = 50 \cdot \ln(3)$
   • $\ln(4^{40}) = 40 \cdot \ln(4)$
   • $\ln(5^{30}) = 30 \cdot \ln(5)$
   • $\ln(6^{20}) = 20 \cdot \ln(6)$

2. Use approximate logarithm values:

   • $\ln(3) \approx 1.0986$
   • $\ln(4) = 2 \cdot \ln(2) \approx 1.3863$
   • $\ln(5) \approx 1.6094$
   • $\ln(6) \approx 1.7918$

3. Now, compute each:

   • $50 \cdot \ln(3) \approx 50 \cdot 1.0986 = 54.93$
   • $40 \cdot \ln(4) \approx 40 \cdot 1.3863 = 55.45$
   • $30 \cdot \ln(5) \approx 30 \cdot 1.6094 = 48.28$
   • $20 \cdot \ln(6) \approx 20 \cdot 1.7918 = 35.84$

Conclusion:

The largest value comes from $\ln(4^{40}) \approx 55.45$, so $4^{40}$ is the largest number.

Let me know if you need further clarification or details on this method!

Related Questions:

1. What would happen if the powers of these numbers were doubled?
2. How does the growth of exponential functions compare to polynomial functions?
3. Can you compare the values of $7^{15}$ and $8^{12}$?
4. What are the properties of logarithms that make comparisons like these easier?
5. How do the base numbers affect the comparison if we change them slightly?

Tip: When comparing large exponential numbers, logarithms simplify the problem by turning multiplication into addition!